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class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> </ul> </nav> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2402.12286">arXiv:2402.12286</a>  [<a href="https://arxiv.org/pdf/2402.12286">pdf</a>, <a href="https://arxiv.org/format/2402.12286">other</a>]  <div class="tags is-inline-block"> math.GT </div> </div> Character varieties of torus links Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+J">Javier Mart铆nez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: In this paper, we study the geometry of the moduli space of representations of the fundamental group of the complement of a torus link into an algebraic group G, an algebraic variety known as the G-character variety of the torus link. These torus links are a family of links in the 3-dimensional sphere formed by stacking several copies of torus knots. We develop an intrinsic stratification of the v… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12286v1-abstract-full').style.display = 'inline'; document.getElementById('2402.12286v1-abstract-short').style.display = 'none';">▽ More</a> In this paper, we study the geometry of the moduli space of representations of the fundamental group of the complement of a torus link into an algebraic group G, an algebraic variety known as the G-character variety of the torus link. These torus links are a family of links in the 3-dimensional sphere formed by stacking several copies of torus knots. We develop an intrinsic stratification of the variety that allows us to relate its geometry with the one of the underlying torus knot. Using this information, we explicitly compute the E-polynomial associated to the Hodge structure of these varieties for $G=SL_2(\mathbb{C})$ and $SL_3(\mathbb{C})$, for an arbitrary torus link, showing an unexpected relation with the number of strands of the link. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.12286v1-abstract-full').style.display = 'none'; document.getElementById('2402.12286v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 19 February, 2024; originally announced February 2024. Comments: 30 pages, 1 figure MSC Class: 57K31; 14D20; 14C30 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2402.02660">arXiv:2402.02660</a>  [<a href="https://arxiv.org/pdf/2402.02660">pdf</a>, <a href="https://arxiv.org/ps/2402.02660">ps</a>, <a href="https://arxiv.org/format/2402.02660">other</a>]  <div class="tags is-inline-block"> math.NT math-ph math.CA math.CV </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.4310/CNTP.240904005850">10.4310/CNTP.240904005850 </a> </div> </div> </div> Stirling-Ramanujan constants are exponential periods Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Perez-Marco%2C+R">Ricardo Perez-Marco</a> Abstract: Ramanujan studied a general class of Stirling constants that are the resummation of some natural divergent series. These constants include the classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find natural integral representations for all these constants that appear as exponential periods in the field $\mathbb Q (t,e^{-t})$ which reveals their natural transalgebraic nature.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02660v3-abstract-full').style.display = 'inline'; document.getElementById('2402.02660v3-abstract-short').style.display = 'none';">▽ More</a> Ramanujan studied a general class of Stirling constants that are the resummation of some natural divergent series. These constants include the classical Euler-Mascheroni, Stirling and Glaisher-Kinkelin constants. We find natural integral representations for all these constants that appear as exponential periods in the field $\mathbb Q (t,e^{-t})$ which reveals their natural transalgebraic nature. We conjecture that all these constants are transcendental numbers. Euler-Mascheroni's and Stirling's integral formula are classical, but the integral formula for Glaisher-Kinkelin appears to be new, as well as the integral formulas for the higher Stirling-Ramanujan constants. The method presented generalizes naturally to prove that many other constants are exponential periods over the field $\mathbb Q(t,e^{-t})$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02660v3-abstract-full').style.display = 'none'; document.getElementById('2402.02660v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 25 September, 2024; v1 submitted 4 February, 2024; originally announced February 2024. Comments: 20 pages. Final published version MSC Class: 40A05; 40G99; 11J81; 33B15; 11B68 Journal ref: Communications in Number Theory and Physics, Volume 18, Number 3, 485-507, 2024 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2311.07572">arXiv:2311.07572</a>  [<a href="https://arxiv.org/pdf/2311.07572">pdf</a>, <a href="https://arxiv.org/ps/2311.07572">ps</a>, <a href="https://arxiv.org/format/2311.07572">other</a>]  <div class="tags is-inline-block"> math.DG </div> </div> The local moduli space of the Einstein-Yang-Mills system Authors: <a href="/search/math?searchtype=author&query=Bunk%2C+S">Severin Bunk</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Shahbazi%2C+C+S">C. S. Shahbazi</a> Abstract: We study the deformation theory of the Einstein-Yang-Mills system on a principal bundle with a compact structure group over a compact manifold. We first construct, as an application of the general slice theorem of Diez and Rudolph, a smooth slice in the tame Fr茅chet category for the coupled action of bundle automorphisms on metrics and connections. Using this result, together with a careful analys… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.07572v1-abstract-full').style.display = 'inline'; document.getElementById('2311.07572v1-abstract-short').style.display = 'none';">▽ More</a> We study the deformation theory of the Einstein-Yang-Mills system on a principal bundle with a compact structure group over a compact manifold. We first construct, as an application of the general slice theorem of Diez and Rudolph, a smooth slice in the tame Fr茅chet category for the coupled action of bundle automorphisms on metrics and connections. Using this result, together with a careful analysis of the linearization of the Einstein-Yang-Mills system, we realize the moduli space of Einstein-Yang-Mills pairs modulo automorphism as an analytic set in a finite-dimensional tame Fr茅chet manifold, extending classical results of Koiso for Einstein metrics and Yang-Mills connections to the Einstein-Yang-Mills system. Furthermore, we introduce the notion of \emph{essential deformation} of an Einstein-Yang-Mills pair, which we characterize in full generality and explore in more detail in the four-dimensional case, proving a decoupling result for trace deformations when the underlying Einstein-Yang-Mills pair is a Ricci-flat metric coupled to an anti-self-dual instanton. In particular, we find a novel obstruction that does not occur in the separate Einstein or Yang-Mills moduli problems. Finally, we prove that every essential deformation of the four-dimensional Einstein-Yang-Mills system based on a Calabi-Yau metric coupled to an instanton is of restricted type. Notable examples of Einstein-Yang-Mills pairs include $\mathrm{G}_2$ or $\mathrm{Spin}(7)$ holonomy metrics coupled to $\mathrm{G}_2$ or $\mathrm{Spin}(7)$ instantons, respectively, or zero-slope polystable holomorphic vector bundles on Calabi-Yau manifolds. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.07572v1-abstract-full').style.display = 'none'; document.getElementById('2311.07572v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 13 November, 2023; originally announced November 2023. Comments: 39 pages </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2303.06218">arXiv:2303.06218</a>  [<a href="https://arxiv.org/pdf/2303.06218">pdf</a>, <a href="https://arxiv.org/format/2303.06218">other</a>]  <div class="tags is-inline-block"> math.GT math.AG </div> </div> Stratification of $\mathrm{SU}(r)$-character varieties of twisted Hopf links Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+J">Javier Mart铆nez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $螕_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometri… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.06218v1-abstract-full').style.display = 'inline'; document.getElementById('2303.06218v1-abstract-short').style.display = 'none';">▽ More</a> We describe the geometry of the character variety of representations of the fundamental group of the complement of a Hopf link with $n$ twists, namely $螕_{n}=\langle x,y \,| \, [x^n,y]=1 \rangle$ into the group $\mathrm{SU}(r)$. For arbitrary rank, we provide geometric descriptions of the loci of irreducible and totally reducible representations. In the case $r = 2$, we provide a complete geometric description of the character variety, proving that this $\mathrm{SU}(2)$-character variety is a deformation retract of the larger $\mathrm{SL}(2,\mathbb{C})$-character variety, as conjectured by Florentino and Lawton. In the case $r = 3$, we also describe different strata of the $\mathrm{SU}(3)$-character variety according to the semi-simple type of the representation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2303.06218v1-abstract-full').style.display = 'none'; document.getElementById('2303.06218v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 10 March, 2023; originally announced March 2023. Comments: Dedicated to Peter E. Newstead, on his $80^{\text{th}}$ birthday. 19 pages, 3 figures MSC Class: 57K31; 14D20; 14M35 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2301.13719">arXiv:2301.13719</a>  [<a href="https://arxiv.org/pdf/2301.13719">pdf</a>, <a href="https://arxiv.org/ps/2301.13719">ps</a>, <a href="https://arxiv.org/format/2301.13719">other</a>]  <div class="tags is-inline-block"> math.GT math.AT math.NT </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1016/j.aim.2024.109942">10.1016/j.aim.2024.109942 </a> </div> </div> </div> Finite sets containing zero are mapping degree sets Authors: <a href="/search/math?searchtype=author&query=Costoya%2C+C">Cristina Costoya</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Viruel%2C+A">Antonio Viruel</a> Abstract: In this paper we solve in the positive the question of whether any finite set of integers, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the rational setting, where an affirmative answer is also given. In this paper we solve in the positive the question of whether any finite set of integers, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the rational setting, where an affirmative answer is also given. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2301.13719v5-abstract-full').style.display = 'none'; document.getElementById('2301.13719v5-abstract-short').style.display = 'inline';">△ Less</a> Submitted 2 September, 2024; v1 submitted 31 January, 2023; originally announced January 2023. Comments: Final version accepted for publication in Advances in Mathematics MSC Class: 55M25; 57N65; 55P62; 55R10 Journal ref: Advances in Mathematics, Volume 457, 2024, 109942 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2208.13046">arXiv:2208.13046</a>  [<a href="https://arxiv.org/pdf/2208.13046">pdf</a>, <a href="https://arxiv.org/ps/2208.13046">ps</a>, <a href="https://arxiv.org/format/2208.13046">other</a>]  <div class="tags is-inline-block"> math.DG math.AT </div> </div> Nearly parallel $G_2$-manifolds: formality and associative submanifolds Authors: <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=Fino%2C+A">Anna Fino</a>, <a href="/search/math?searchtype=author&query=Kovalev%2C+A">Alexei Kovalev</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We construct new examples of non-formal simply connected compact Sasaki-Einstein 7-manifolds. We determine the minimal model of the total space of any fibre bundle over $CP^2$ with fibre $S^1\times S^2$ or $S^3/Z_p$ ($p>0$), and we apply this to conclude that the Aloff-Wallach spaces are formal. We also find examples of formal manifolds and non-formal manifolds, which are locally conformal paralle… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.13046v3-abstract-full').style.display = 'inline'; document.getElementById('2208.13046v3-abstract-short').style.display = 'none';">▽ More</a> We construct new examples of non-formal simply connected compact Sasaki-Einstein 7-manifolds. We determine the minimal model of the total space of any fibre bundle over $CP^2$ with fibre $S^1\times S^2$ or $S^3/Z_p$ ($p>0$), and we apply this to conclude that the Aloff-Wallach spaces are formal. We also find examples of formal manifolds and non-formal manifolds, which are locally conformal parallel $Spin(7)$-manifolds. On the other hand, we construct associative minimal submanifolds in the Aloff-Wallach spaces and in any regular Sasaki-Einstein 7-manifold; in particular, in the space $Q(1,1,1)=(SU(2) \times SU(2) \times SU(2))/ (U(1) \times U(1))$ with the natural $S^1$-family of nearly parallel $G_2$-structures induced by the Sasaki-Einstein structure. In each of those cases, we obtain a family of non-trivial associative deformations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2208.13046v3-abstract-full').style.display = 'none'; document.getElementById('2208.13046v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 18 April, 2023; v1 submitted 27 August, 2022; originally announced August 2022. Comments: 35 pages, no figures; v2. added references; v3. title changed, clarifications in the Introduction and in the proof of Theorem 4.2, rewritten some parts to improve presentation MSC Class: 53C25; 53C30; 53B25; 53C40 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2207.09170">arXiv:2207.09170</a>  [<a href="https://arxiv.org/pdf/2207.09170">pdf</a>, <a href="https://arxiv.org/format/2207.09170">other</a>]  <div class="tags is-inline-block"> math.GT math.AG </div> </div> Geometry of $\mathrm{SU}(3)$-character varieties of torus knots Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mart%C3%ADnez%2C+J">Javier Mart铆nez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We describe the geometry of the character variety of representations of the knot group $螕_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally reducible representations, representations decomposing into a $2$-dimensional and a $1$-dimensional representation, and irreducible representations, the latter of two ty… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.09170v1-abstract-full').style.display = 'inline'; document.getElementById('2207.09170v1-abstract-short').style.display = 'none';">▽ More</a> We describe the geometry of the character variety of representations of the knot group $螕_{m,n}=\langle x,y| x^n=y^m\rangle$ into the group $\mathrm{SU}(3)$, by stratifying the character variety into strata correspoding to totally reducible representations, representations decomposing into a $2$-dimensional and a $1$-dimensional representation, and irreducible representations, the latter of two types depending on whether the matrices have distinct eigenvalues, or one of the matrices has one eigenvalue of multiplicity $2$. We describe how the closure of each stratum meets lower strata, and use this to compute the compactly supported Euler characteristic, and to prove that the inclusion of the character variety for $\mathrm{SU}(3)$ into the character variety for $\mathrm{SL}(3,\mathbb{C})$ is a homotopy equivalence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.09170v1-abstract-full').style.display = 'none'; document.getElementById('2207.09170v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 19 July, 2022; originally announced July 2022. Comments: 22 pages, 4 figures MSC Class: 57K31; 14D20; 14M35 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2207.01335">arXiv:2207.01335</a>  [<a href="https://arxiv.org/pdf/2207.01335">pdf</a>, <a href="https://arxiv.org/ps/2207.01335">ps</a>, <a href="https://arxiv.org/format/2207.01335">other</a>]  <div class="tags is-inline-block"> math.RA </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1007/s13398-023-01414-w">10.1007/s13398-023-01414-w </a> </div> </div> </div> Automorphism groups of Cayley evolution algebras Authors: <a href="/search/math?searchtype=author&query=Costoya%2C+C">Cristina Costoya</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Tocino%2C+A">Alicia Tocino</a>, <a href="/search/math?searchtype=author&query=Viruel%2C+A">Antonio Viruel</a> Abstract: In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra. In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is isomorphic to $Aut(X)$ where $X$ is a finite-dimensional absolutely simple Cayley evolution $k$-algebra. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2207.01335v2-abstract-full').style.display = 'none'; document.getElementById('2207.01335v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 9 March, 2023; v1 submitted 4 July, 2022; originally announced July 2022. Comments: This version of the article has been accepted for publication, after peer review but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is published under the Creative Commons Attribution license and can be freely dowloaded it from https://doi.org/10.1007/s13398-023-01414-w MSC Class: 05C25; 17A36; 17D99 Journal ref: Rev. R. Acad. Cienc. Exactas F铆s. Nat. Ser. A Mat. RACSAM 117, 82 (2023) </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2202.07227">arXiv:2202.07227</a>  [<a href="https://arxiv.org/pdf/2202.07227">pdf</a>, <a href="https://arxiv.org/ps/2202.07227">ps</a>, <a href="https://arxiv.org/format/2202.07227">other</a>]  <div class="tags is-inline-block"> math.AG math.RT </div> </div> Coordinate rings of some $SL_2$-character varieties Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Ovejero%2C+J+M">Jes煤s Mart铆n Ovejero</a> Abstract: We determine generators of the coordinate ring of $SL_2$-character varieties. In the case of the free group $F_3$ we obtain an explicit equation of the $SL_2$-character variety. For free groups $F_k$ we find transcendental generators. Finally, for the case of the 2-torus, we get an explicit equation of the $SL_2$-character variety and use the description to compute their E-polynomials. We determine generators of the coordinate ring of $SL_2$-character varieties. In the case of the free group $F_3$ we obtain an explicit equation of the $SL_2$-character variety. For free groups $F_k$ we find transcendental generators. Finally, for the case of the 2-torus, we get an explicit equation of the $SL_2$-character variety and use the description to compute their E-polynomials. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.07227v2-abstract-full').style.display = 'none'; document.getElementById('2202.07227v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 11 April, 2022; v1 submitted 15 February, 2022; originally announced February 2022. Comments: 16 pages, no figures; v2. References added MSC Class: 14M35; 20G05; 14D20 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2202.07090">arXiv:2202.07090</a>  [<a href="https://arxiv.org/pdf/2202.07090">pdf</a>, <a href="https://arxiv.org/format/2202.07090">other</a>]  <div class="tags is-inline-block"> math.GT math.AG </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1007/s00009-023-02300-w">10.1007/s00009-023-02300-w </a> </div> </div> </div> Representation varieties of twisted Hopf links Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representati… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.07090v2-abstract-full').style.display = 'inline'; document.getElementById('2202.07090v2-abstract-short').style.display = 'none';">▽ More</a> We study the representation theory of the fundamental group of the complement of a Hopf link with n twists. A general framework is described to analyze the $SL_r(C)$-representation varieties of these twisted Hopf links as byproduct of a combinatorial problem and equivariant Hodge theory. As application, close formulas of their E-polynomials are provided for ranks 2 and 3, both for the representation and character varieties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.07090v2-abstract-full').style.display = 'none'; document.getElementById('2202.07090v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 19 February, 2024; v1 submitted 14 February, 2022; originally announced February 2022. Comments: 25 pages, 3 figures; v2. Corrected mistake in the E-polynomial in main theorem MSC Class: 57K31; 14D20; 14C30 Journal ref: Mediterranean Journal of Mathematics volume 20, 2023 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2201.09362">arXiv:2201.09362</a>  [<a href="https://arxiv.org/pdf/2201.09362">pdf</a>, <a href="https://arxiv.org/ps/2201.09362">ps</a>, <a href="https://arxiv.org/format/2201.09362">other</a>]  <div class="tags is-inline-block"> math.SG math.AT math.DG </div> </div> Asymptotically holomorphic theory for symplectic orbifolds Authors: <a href="/search/math?searchtype=author&query=Gironella%2C+F">Fabio Gironella</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Zhou%2C+Z">Zhengyi Zhou</a> Abstract: We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these s… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.09362v2-abstract-full').style.display = 'inline'; document.getElementById('2201.09362v2-abstract-short').style.display = 'none';">▽ More</a> We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient manifold satisfies those properties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2201.09362v2-abstract-full').style.display = 'none'; document.getElementById('2201.09362v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 18 February, 2022; v1 submitted 23 January, 2022; originally announced January 2022. Comments: 30 pages, no figures; v2. References added </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2111.08399">arXiv:2111.08399</a>  [<a href="https://arxiv.org/pdf/2111.08399">pdf</a>, <a href="https://arxiv.org/ps/2111.08399">ps</a>, <a href="https://arxiv.org/format/2111.08399">other</a>]  <div class="tags is-inline-block"> math.DG math.AT </div> </div> Purely coclosed $G_2$-structures on nilmanifolds Authors: <a href="/search/math?searchtype=author&query=Bazzoni%2C+G">Giovanni Bazzoni</a>, <a href="/search/math?searchtype=author&query=Garv%C3%ADn%2C+A">Antonio Garv铆n</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We classify 7-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed $G_2$-structures. This is done by going through the list of all 7-dimensional nilpotent Lie algebras given by Gong, providing an example of a left-invariant 3-form $\varphi$ which is a pure coclosed $G_2$-structure (that is, it satisfies $d*\varphi=0$,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.08399v1-abstract-full').style.display = 'inline'; document.getElementById('2111.08399v1-abstract-short').style.display = 'none';">▽ More</a> We classify 7-dimensional nilpotent Lie groups, decomposable or of nilpotency step at most 4, endowed with left-invariant purely coclosed $G_2$-structures. This is done by going through the list of all 7-dimensional nilpotent Lie algebras given by Gong, providing an example of a left-invariant 3-form $\varphi$ which is a pure coclosed $G_2$-structure (that is, it satisfies $d*\varphi=0$, $\varphi \wedge d\varphi=0$) for those nilpotent Lie algebras that admit them; and by showing the impossibility of having a purely coclosed $G_2$-structure for the rest of them. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.08399v1-abstract-full').style.display = 'none'; document.getElementById('2111.08399v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 16 November, 2021; originally announced November 2021. Comments: 26 pages, 16 tables, Accompanying SageMath worksheets available at http://agt.cie.uma.es/~vicente.munoz/Sage-worksheets.zip MSC Class: 53C15; 22E25; 53C38; 17B30 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2105.08945">arXiv:2105.08945</a>  [<a href="https://arxiv.org/pdf/2105.08945">pdf</a>, <a href="https://arxiv.org/format/2105.08945">other</a>]  <div class="tags is-inline-block"> math.AG </div> </div> The point counting problem in representation varieties of torus knots Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interp… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.08945v2-abstract-full').style.display = 'inline'; document.getElementById('2105.08945v2-abstract-short').style.display = 'none';">▽ More</a> We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1$ and $AGL_2$ for an arbitrary field $k$. In the case that $k = F_q$ is a finite field this gives rise to the count of the number of points of the representation variety, while for $k = C$ this calculation returns the E-polynomial of the representation variety. We discuss the interplay between these two results in sight of Katz theorem that relates the point count polynomial with the E-polynomial. In particular, we shall show that several point count polynomials exist for these representation varieties, depending on the arithmetic between m,n and the characteristic of the field, whereas only one of them agrees with the actual E-polynomial. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.08945v2-abstract-full').style.display = 'none'; document.getElementById('2105.08945v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 22 June, 2021; v1 submitted 19 May, 2021; originally announced May 2021. Comments: 20 pages, 4 figures. arXiv admin note: text overlap with arXiv:2104.13651 MSC Class: 14G15; 14D20; 20G15; 14C30 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2104.13651">arXiv:2104.13651</a>  [<a href="https://arxiv.org/pdf/2104.13651">pdf</a>, <a href="https://arxiv.org/ps/2104.13651">ps</a>, <a href="https://arxiv.org/format/2104.13651">other</a>]  <div class="tags is-inline-block"> math.AG </div> </div> Motive of the representation varieties of torus knots for low rank affine groups Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C]. We compute the motive of the variety of representations of the torus knot of type (m,n) into the affine groups $AGL_1(C)$ and $AGL_2(C)$. For this, we stratify the varieties and show that the motives lie in the subring generated by the Lefschetz motive q=[C]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2104.13651v1-abstract-full').style.display = 'none'; document.getElementById('2104.13651v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 28 April, 2021; originally announced April 2021. Comments: 13 pages, no figures MSC Class: 14C30; 57R56; 14L24; 14D21 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2101.09103">arXiv:2101.09103</a>  [<a href="https://arxiv.org/pdf/2101.09103">pdf</a>, <a href="https://arxiv.org/format/2101.09103">other</a>]  <div class="tags is-inline-block"> math.CO cs.GT </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1142/S0219198920500206">10.1142/S0219198920500206 </a> </div> </div> </div> Nash Equilibria in certain two-choice multi-player games played on the ladder graph Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V+S">Victoria S谩nchez Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Gettrick%2C+M+M">Michael Mc Gettrick</a> Abstract: In this article we compute analytically the number of Nash Equilibria (NE) for a two-choice game played on a (circular) ladder graph with $2n$ players. We consider a set of games with generic payoff parameters, with the only requirement that a NE occurs if the players choose opposite strategies (anti-coordination game). The results show that for both, the ladder and circular ladder, the number of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.09103v1-abstract-full').style.display = 'inline'; document.getElementById('2101.09103v1-abstract-short').style.display = 'none';">▽ More</a> In this article we compute analytically the number of Nash Equilibria (NE) for a two-choice game played on a (circular) ladder graph with $2n$ players. We consider a set of games with generic payoff parameters, with the only requirement that a NE occurs if the players choose opposite strategies (anti-coordination game). The results show that for both, the ladder and circular ladder, the number of NE grows exponentially with (half) the number of players $n$, as $N_{NE}(2n)\sim C(\varphi)^n$, where $\varphi=1.618..$ is the golden ratio and $C_{circ}>C_{ladder}$. In addition, the value of the scaling factor $C_{ladder}$ depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is $C_{circ}$ is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.09103v1-abstract-full').style.display = 'none'; document.getElementById('2101.09103v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 22 January, 2021; originally announced January 2021. Comments: 15 pages, 7 figures. Paper online ready in journal International Game Theory Review MSC Class: 91A43; 05C57 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2101.01961">arXiv:2101.01961</a>  [<a href="https://arxiv.org/pdf/2101.01961">pdf</a>, <a href="https://arxiv.org/ps/2101.01961">ps</a>, <a href="https://arxiv.org/format/2101.01961">other</a>]  <div class="tags is-inline-block"> math.GT math.AT </div> </div> On strongly inflexible manifolds Authors: <a href="/search/math?searchtype=author&query=Costoya%2C+C">Cristina Costoya</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Viruel%2C+A">Antonio Viruel</a> Abstract: An oriented closed connected N-manifold M is inflexible if it does not admit self-maps of unbounded degree. In addition, if all the maps from any other oriented closed connected N-manifold have bounded degree, then M is said to be strongly inflexible. The existence of simply-connected inflexible manifolds was established by Arkowitz and Lupton. However, the existence of simply-connected strongly i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.01961v4-abstract-full').style.display = 'inline'; document.getElementById('2101.01961v4-abstract-short').style.display = 'none';">▽ More</a> An oriented closed connected N-manifold M is inflexible if it does not admit self-maps of unbounded degree. In addition, if all the maps from any other oriented closed connected N-manifold have bounded degree, then M is said to be strongly inflexible. The existence of simply-connected inflexible manifolds was established by Arkowitz and Lupton. However, the existence of simply-connected strongly inflexible manifolds is still an open question. We provide an algorithm relying on Sullivan models that allow us to prove that all, but one, of the known examples of simply-connected inflexible manifolds are not strongly inflexible. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.01961v4-abstract-full').style.display = 'none'; document.getElementById('2101.01961v4-abstract-short').style.display = 'inline';">△ Less</a> Submitted 7 February, 2022; v1 submitted 6 January, 2021; originally announced January 2021. Comments: 26 pages, no figures; v2. overall improved version; v3. fully revised version MSC Class: 55P62; 57N65; 55P10 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2011.05783">arXiv:2011.05783</a>  [<a href="https://arxiv.org/pdf/2011.05783">pdf</a>, <a href="https://arxiv.org/format/2011.05783">other</a>]  <div class="tags is-inline-block"> math.DG math.AG math.SG </div> </div> A Smale-Barden manifold admitting K-contact but not Sasakian structure Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We give the first example of a simply connected compact 5-manifold (Smale-Barden manifold) which admits a K-contact structure but does not admit any Sasakian structure, settling a long standing question of Boyer and Galicki. We give the first example of a simply connected compact 5-manifold (Smale-Barden manifold) which admits a K-contact structure but does not admit any Sasakian structure, settling a long standing question of Boyer and Galicki. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2011.05783v2-abstract-full').style.display = 'none'; document.getElementById('2011.05783v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 12 July, 2023; v1 submitted 11 November, 2020; originally announced November 2020. Comments: 33 pages, 2 figures; version 2 largely improved, a mistake corrected, and the arguments simplified MSC Class: 57R18; 53C25; 53D35; 57R17 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2007.08597">arXiv:2007.08597</a>  [<a href="https://arxiv.org/pdf/2007.08597">pdf</a>, <a href="https://arxiv.org/ps/2007.08597">ps</a>, <a href="https://arxiv.org/format/2007.08597">other</a>]  <div class="tags is-inline-block"> math.DG math.AG math.SG </div> </div> Negative Sasakian structures on simply-connected 5-manifolds Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">V. Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Sch%C3%BCtt%2C+M">M. Sch眉tt</a>, <a href="/search/math?searchtype=author&query=Tralle%2C+A">A. Tralle</a> Abstract: We study several questions on the existence of negative Sasakian structures on simply connected rational homology spheres and on Smale-Barden manifolds of the form $\#_k(S^2\times S^3)$. First, we prove that any simply connected rational homology sphere admitting positive Sasakian structures also admits a negative one. This result answers the question, posed by Boyer and Galicki in their book [BG]… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.08597v1-abstract-full').style.display = 'inline'; document.getElementById('2007.08597v1-abstract-short').style.display = 'none';">▽ More</a> We study several questions on the existence of negative Sasakian structures on simply connected rational homology spheres and on Smale-Barden manifolds of the form $\#_k(S^2\times S^3)$. First, we prove that any simply connected rational homology sphere admitting positive Sasakian structures also admits a negative one. This result answers the question, posed by Boyer and Galicki in their book [BG], of determining which simply connected rational homology spheres admit both negative and positive Sasakian structures. Second, we prove that the connected sum $\#_k(S^2\times S^3)$ admits negative quasi-regular Sasakian structures for any $k$. This yields a complete answer to another question posed in [BG]. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.08597v1-abstract-full').style.display = 'none'; document.getElementById('2007.08597v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 16 July, 2020; originally announced July 2020. Comments: 24 pages </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2006.01810">arXiv:2006.01810</a>  [<a href="https://arxiv.org/pdf/2006.01810">pdf</a>, <a href="https://arxiv.org/ps/2006.01810">ps</a>, <a href="https://arxiv.org/format/2006.01810">other</a>]  <div class="tags is-inline-block"> math.AG math.GT </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1016/j.jalgebra.2022.06.008">10.1016/j.jalgebra.2022.06.008 </a> </div> </div> </div> Motive of the $SL_4$-character variety of torus knots Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+A">Angel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$ of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.01810v3-abstract-full').style.display = 'inline'; document.getElementById('2006.01810v3-abstract-short').style.display = 'none';">▽ More</a> In this paper, we compute the motive of the character variety of representations of the fundamental group of the complement of an arbitrary torus knot into $SL_4(k)$, for any algebraically closed field $k$ of zero characteristic. For that purpose, we introduce a stratification of the variety in terms of the type of a canonical filtration attached to any representation. This allows us to reduce the computation of the motive to a combinatorial problem. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2006.01810v3-abstract-full').style.display = 'none'; document.getElementById('2006.01810v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 14 May, 2022; v1 submitted 2 June, 2020; originally announced June 2020. Comments: 37 pages. Note: download source for the output of all the strata as separate file MSC Class: 14D20; 57M25; 57M27 Journal ref: Journal of Algebra Volume 610, 2022 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2005.01841">arXiv:2005.01841</a>  [<a href="https://arxiv.org/pdf/2005.01841">pdf</a>, <a href="https://arxiv.org/format/2005.01841">other</a>]  <div class="tags is-inline-block"> math.AG </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1007/978-3-030-84721-0_18">10.1007/978-3-030-84721-0_18 </a> </div> </div> </div> Representation variety for the rank one affine group Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+A">Angel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which pe… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.01841v3-abstract-full').style.display = 'inline'; document.getElementById('2005.01841v3-abstract-short').style.display = 'none';">▽ More</a> The aim of this paper is to study the virtual classes of representation varieties of surface groups onto the rank one affine group. We perform this calculation by three different approaches: the geometric method, based on stratifying the representation variety into simpler pieces; the arithmetic method, focused on counting their number of points over finite fields; and the quantum method, which performs the computation by means of a Topological Quantum Field Theory. We also discuss the corresponding moduli spaces of representations and character varieties, which turn out to be non-equivalent due to the non-reductiveness of the underlying group. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.01841v3-abstract-full').style.display = 'none'; document.getElementById('2005.01841v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 11 September, 2020; v1 submitted 4 May, 2020; originally announced May 2020. Comments: 28 pages, 3 figures. References added MSC Class: 57R56; 14C30; 14D07; 14D21 Journal ref: Mathematical Analysis in Interdisciplinary Research. Springer Optimization and Its Applications, vol 179, 2021 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2004.12643">arXiv:2004.12643</a>  [<a href="https://arxiv.org/pdf/2004.12643">pdf</a>, <a href="https://arxiv.org/format/2004.12643">other</a>]  <div class="tags is-inline-block"> math.DG math.AG math.SG </div> </div> Quasi-regular Sasakian and K-contact structures on Smale-Barden manifolds Authors: <a href="/search/math?searchtype=author&query=Ca%C3%B1as%2C+A">A. Ca帽as</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">V. Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Sch%C3%BCtt%2C+M">M. Sch眉tt</a>, <a href="/search/math?searchtype=author&query=Tralle%2C+A">A. Tralle</a> Abstract: Smale-Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale-Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Koll谩r).… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.12643v1-abstract-full').style.display = 'inline'; document.getElementById('2004.12643v1-abstract-short').style.display = 'none';">▽ More</a> Smale-Barden manifolds are simply-connected closed 5-manifolds. It is an important and difficult question to decide when a Smale-Barden manifold admits a Sasakian or a K-contact structure. The known constructions of Sasakian and K-contact structures are obtained mainly by two techniques. These are either links (Boyer and Galicki), or semi-regular Seifert fibrations over smooth orbifolds (Koll谩r). Recently, the second named author of this article started the systematic development of quasi-regular Seifert fibrations, that is, over orbifolds which are not necessarily smooth. The present work is devoted to several applications of this theory. First, we develop constructions of a Smale-Barden manifold admitting a quasi-regular Sasakian structure but not a semi-regular K-contact structure. Second, we determine all Smale-Barden manifolds that admit a null Sasakian structure. Finally, we show a counterexample in the realm of cyclic K盲hler orbifolds to the algebro-geometric conjecture that claims that for an algebraic surface with $b_1=0$ and $b_2>1$ there cannot be $b_2$ smooth disjoint complex curves of genus g>0 spanning the (rational) homology. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.12643v1-abstract-full').style.display = 'none'; document.getElementById('2004.12643v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 27 April, 2020; originally announced April 2020. Comments: 23 pages, 1 figure MSC Class: 53C25; 53D35; 14J28; 14J17 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2003.07319">arXiv:2003.07319</a>  [<a href="https://arxiv.org/pdf/2003.07319">pdf</a>, <a href="https://arxiv.org/format/2003.07319">other</a>]  <div class="tags is-inline-block"> math.DG math.SG </div> </div> Gompf connected sum for orbifolds and K-contact Smale-Barden manifolds Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with $b_1=0$ and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale-Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estim… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.07319v1-abstract-full').style.display = 'inline'; document.getElementById('2003.07319v1-abstract-short').style.display = 'none';">▽ More</a> We develop the Gompf fiber connected sum operation for symplectic orbifolds. We use it to construct a symplectic 4-orbifold with $b_1=0$ and containing symplectic surfaces of genus 1 and 2 that are disjoint and span the rational homology. This is used in turn to construct a K-contact Smale-Barden manifold with specified 2-homology that satisfies the known topological constraints with sharper estimates than the examples constructed previously. The manifold can be chosen spin or non-spin. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2003.07319v1-abstract-full').style.display = 'none'; document.getElementById('2003.07319v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 16 March, 2020; originally announced March 2020. Comments: 35 pages, 2 figures MSC Class: 57R18; 53C25; 53D35; 57R17 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/2002.00457">arXiv:2002.00457</a>  [<a href="https://arxiv.org/pdf/2002.00457">pdf</a>, <a href="https://arxiv.org/ps/2002.00457">ps</a>, <a href="https://arxiv.org/format/2002.00457">other</a>]  <div class="tags is-inline-block"> math.DG math.AG </div> </div> On the classification of Smale-Barden manifolds with Sasakian structures Authors: <a href="/search/math?searchtype=author&query=Tralle%2C+A">Aleksy Tralle</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: Smale-Barden manifolds $M$ are classified by their second homology $H_2(M,{\mathbb Z})$ and the Barden invariant $i(M)$. It is an important and dificult question to decide when $M$ admits a Sasakian structure in terms of these data. In this work we show methods of doing this. In particular we realize all $M$ with $H_2(M)={\mathbb Z}^k\oplus(\oplus_{i=1}^r{\mathbb Z}_{m_i}^{2g_i})$ and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.00457v1-abstract-full').style.display = 'inline'; document.getElementById('2002.00457v1-abstract-short').style.display = 'none';">▽ More</a> Smale-Barden manifolds $M$ are classified by their second homology $H_2(M,{\mathbb Z})$ and the Barden invariant $i(M)$. It is an important and dificult question to decide when $M$ admits a Sasakian structure in terms of these data. In this work we show methods of doing this. In particular we realize all $M$ with $H_2(M)={\mathbb Z}^k\oplus(\oplus_{i=1}^r{\mathbb Z}_{m_i}^{2g_i})$ and $i=0,\infty$, provided that $k\geq 1$, $m_i\geq 2$, $g_i\geq 1$, $m_i$ are pairwise coprime. Using our methods we also contribute to the problem of the existence of definite Sasakian structures on rational homology spheres. Also, we give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2002.00457v1-abstract-full').style.display = 'none'; document.getElementById('2002.00457v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 2 February, 2020; originally announced February 2020. Comments: 16 pages, no figures MSC Class: 53C25; 53D35; 57R17; 14J25 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1911.08901">arXiv:1911.08901</a>  [<a href="https://arxiv.org/pdf/1911.08901">pdf</a>, <a href="https://arxiv.org/ps/1911.08901">ps</a>, <a href="https://arxiv.org/format/1911.08901">other</a>]  <div class="tags is-inline-block"> math.DG math.AG math.SG </div> </div> A K-contact simply connected 5-manifold with no semi-regular Sasakian structure Authors: <a href="/search/math?searchtype=author&query=Ca%C3%B1as%2C+A">Alejandro Ca帽as</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Rojo%2C+J">Juan Rojo</a>, <a href="/search/math?searchtype=author&query=Viruel%2C+A">Antonio Viruel</a> Abstract: We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them but one of genus 1 and the other of genus g>1, (b) to pro… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.08901v4-abstract-full').style.display = 'inline'; document.getElementById('1911.08901v4-abstract-short').style.display = 'none';">▽ More</a> We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the homology, all of them but one of genus 1 and the other of genus g>1, (b) to prove a bound on the second Betti number $b_2$ of an algebraic surface with $b_1=0$ and having disjoint complex curves spanning the homology when all of them but one are of genus 1 and the other of genus g>1. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1911.08901v4-abstract-full').style.display = 'none'; document.getElementById('1911.08901v4-abstract-short').style.display = 'inline';">△ Less</a> Submitted 30 October, 2020; v1 submitted 20 November, 2019; originally announced November 2019. Comments: 32 pages, no figures; v2. added a proof that the symplectic 4-manifold in [MRT] cannot be complex; v3. added computation of second Stiefel-Whitney class; v4. small changes. Title changed MSC Class: 53C25; 53D35; 57R17; 14J25 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1906.09629">arXiv:1906.09629</a>  [<a href="https://arxiv.org/pdf/1906.09629">pdf</a>, <a href="https://arxiv.org/ps/1906.09629">ps</a>, <a href="https://arxiv.org/format/1906.09629">other</a>]  <div class="tags is-inline-block"> math.NT math.CO math.CV </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.4064/aa200619-28-9">10.4064/aa200619-28-9 </a> </div> </div> </div> On the genesis of BBP formulas Authors: <a href="/search/math?searchtype=author&query=Barsky%2C+D">Daniel Barsky</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=P%C3%A9rez-Marco%2C+R">Ricardo P茅rez-Marco</a> Abstract: We present a general procedure to generate infinitely many BBP and BBP-like formulas for the simplest transcendental numbers. This provides some insight and a better understanding into their nature. In particular, we can derive the main known BBP formulas for $蟺$. We can understand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas r… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.09629v4-abstract-full').style.display = 'inline'; document.getElementById('1906.09629v4-abstract-short').style.display = 'none';">▽ More</a> We present a general procedure to generate infinitely many BBP and BBP-like formulas for the simplest transcendental numbers. This provides some insight and a better understanding into their nature. In particular, we can derive the main known BBP formulas for $蟺$. We can understand why many of these formulas are rearrangements of each other. We also understand better where some null BBP formulas representing $0$ come from. We also explain what is the observed relation between some BBP formulas for $\log 2$ and $蟺$, that are obtained by taking real and imaginary parts of a general complex BBP formula. Our methods are elementary, but motivated by transalgebraic considerations, and offer a new way to obtain and to search many new BBP formulas and, conjecturally, to better understand transalgebraic relations between transcendental constants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.09629v4-abstract-full').style.display = 'none'; document.getElementById('1906.09629v4-abstract-short').style.display = 'inline';">△ Less</a> Submitted 13 February, 2023; v1 submitted 23 June, 2019; originally announced June 2019. Comments: 24 pages, no figures. v2: Added one reference. Moved previous section 5 to an appendix. v3: References added. Final version to appear in Acta Arithmetica. v4: Corrects minor mistakes in the published version MSC Class: 11K16; 11J99 Journal ref: Acta Arithmetica 198 (2021), 401-426 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1904.08970">arXiv:1904.08970</a>  [<a href="https://arxiv.org/pdf/1904.08970">pdf</a>, <a href="https://arxiv.org/ps/1904.08970">ps</a>, <a href="https://arxiv.org/format/1904.08970">other</a>]  <div class="tags is-inline-block"> math.AG math.AT </div> </div> Rationally elliptic toric varieties Authors: <a href="/search/math?searchtype=author&query=Biswas%2C+I">Indranil Biswas</a>, <a href="/search/math?searchtype=author&query=Munoz%2C+V">Vicente Munoz</a>, <a href="/search/math?searchtype=author&query=Murillo%2C+A">Aniceto Murillo</a> Abstract: We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described. We give a characterization of all complete smooth toric varieties whose rational homotopy is of elliptic type. All such toric varieties of complex dimension not more than three are explicitly described. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1904.08970v4-abstract-full').style.display = 'none'; document.getElementById('1904.08970v4-abstract-short').style.display = 'inline';">△ Less</a> Submitted 3 February, 2020; v1 submitted 18 April, 2019; originally announced April 2019. Comments: 15 pages, no figures. v2: new references added. v3: proof of formality added, references added MSC Class: 14M25; 55P62; 55Q52; 52B20 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1904.01462">arXiv:1904.01462</a>  [<a href="https://arxiv.org/pdf/1904.01462">pdf</a>, <a href="https://arxiv.org/ps/1904.01462">ps</a>, <a href="https://arxiv.org/format/1904.01462">other</a>]  <div class="tags is-inline-block"> math.DG </div> </div> Spin-harmonic structures and nilmanifolds Authors: <a href="/search/math?searchtype=author&query=Bazzoni%2C+G">Giovanni Bazzoni</a>, <a href="/search/math?searchtype=author&query=Martin-Merchan%2C+L">Lucia Martin-Merchan</a>, <a href="/search/math?searchtype=author&query=Munoz%2C+V">Vicente Munoz</a> Abstract: We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1904.01462v2-abstract-full').style.display = 'inline'; document.getElementById('1904.01462v2-abstract-short').style.display = 'none';">▽ More</a> We introduce spin-harmonic structures, a class of geometric structures on Riemannian manifolds of low dimension which are defined by a harmonic unitary spinor. Such structures are related to SU(2) (dim=4,5), SU(3) (dim=6) and G_2 (dim=7) structures; in dimension 8, a spin-harmonic structure is equivalent to a balanced Spin(7) structure. As an application, we obtain examples of compact 8-manifolds endowed with non-integrable Spin(7) structures of balanced type. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1904.01462v2-abstract-full').style.display = 'none'; document.getElementById('1904.01462v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 14 January, 2022; v1 submitted 2 April, 2019; originally announced April 2019. Comments: 33 pages, no figures. Accepted in Communications in Analysis and Geometry MSC Class: 57N16; 15A66; 53C27; 22E25 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1808.07144">arXiv:1808.07144</a>  [<a href="https://arxiv.org/pdf/1808.07144">pdf</a>, <a href="https://arxiv.org/ps/1808.07144">ps</a>, <a href="https://arxiv.org/format/1808.07144">other</a>]  <div class="tags is-inline-block"> math.DG </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1016/j.aim.2021.107623">10.1016/j.aim.2021.107623 </a> </div> </div> </div> A compact $G_2$-calibrated manifold with first Betti number $b_1=1$ Authors: <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=Fino%2C+A">Anna Fino</a>, <a href="/search/math?searchtype=author&query=Kovalev%2C+A">Alexei Kovalev</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We construct a compact formal 7-manifold with a closed $G_2$-structure and with first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it does not admit any $G_2$-structure such that the holonomy group of the associated metric is a subgroup of $G_2$. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed $G_2$-stru… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.07144v6-abstract-full').style.display = 'inline'; document.getElementById('1808.07144v6-abstract-short').style.display = 'none';">▽ More</a> We construct a compact formal 7-manifold with a closed $G_2$-structure and with first Betti number $b_1=1$, which does not admit any torsion-free $G_2$-structure, that is, it does not admit any $G_2$-structure such that the holonomy group of the associated metric is a subgroup of $G_2$. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed $G_2$-structure and, for each of those 3-tori, we show a 3-dimensional family of non-trivial associative deformations. We also construct a fibration of our 7-manifold over $S^2\times S^1$ with generic fiber a (non-calibrated) coassociative 4-torus and some singular fibers. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.07144v6-abstract-full').style.display = 'none'; document.getElementById('1808.07144v6-abstract-short').style.display = 'inline';">△ Less</a> Submitted 14 September, 2022; v1 submitted 21 August, 2018; originally announced August 2018. Comments: 32 pages, v2 new results on the associative 3-folds, new section on the coassociative 4-folds. v3 corrected a mistake in the proof of Thm.17. v4 corrected the proof of the formality of the resolution (Prop.24). v6: changes made after publication: corrected the calculation of the fixed point sets in eqn.(10) and on p.23, minor simplification before Prop.17, Remark 18 rewritten MSC Class: 53C38; 53C15; 17B30; 22E25 Journal ref: Adv. Math. 381 (2021) </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1805.01160">arXiv:1805.01160</a>  [<a href="https://arxiv.org/pdf/1805.01160">pdf</a>, <a href="https://arxiv.org/ps/1805.01160">ps</a>, <a href="https://arxiv.org/format/1805.01160">other</a>]  <div class="tags is-inline-block"> math.SG math.DG </div> </div> Symplectic resolution of orbifolds with homogeneous isotropy Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Rojo%2C+J+A">Juan Angel Rojo</a> Abstract: We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups. We construct the symplectic resolution of a symplectic orbifold whose isotropy locus consists of disjoint submanifolds with homogeneous isotropy, that is, all its points have the same isotropy groups. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.01160v3-abstract-full').style.display = 'none'; document.getElementById('1805.01160v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 16 October, 2020; v1 submitted 3 May, 2018; originally announced May 2018. Comments: 29 pages, no figures. v2 References updated. To appear in Geometriae Dedicata. v3 Corrected a small technical error of v2 where the singularity of the orbifold symplectic form at the origin was not handled correctly. Some extra minor modifications have been made along the way, mainly notational issues MSC Class: 53D05; 57R17; 57R18 Journal ref: Geometriae Dedicata, Vol. 204, no. 1, 2020, 339-363 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1709.08806">arXiv:1709.08806</a>  [<a href="https://arxiv.org/pdf/1709.08806">pdf</a>, <a href="https://arxiv.org/ps/1709.08806">ps</a>, <a href="https://arxiv.org/format/1709.08806">other</a>]  <div class="tags is-inline-block"> math.DG math.AT </div> </div> On SO(3)-bundles over the Wolf spaces Authors: <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=S%C3%A1nchez%2C+J">Jonatan S谩nchez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We study the formality of the total space of principal SU(2) and SO(3)-bundles over a Wolf space, that is a symmetric positive quaternionic K盲hker manifold. We apply this to conclude that all the 3-Sasakian homogeneous spaces are formal. We also determine the principal SU(2) and SO(3)-bundles over the Wolf spaces whose total space is non-formal. We study the formality of the total space of principal SU(2) and SO(3)-bundles over a Wolf space, that is a symmetric positive quaternionic K盲hker manifold. We apply this to conclude that all the 3-Sasakian homogeneous spaces are formal. We also determine the principal SU(2) and SO(3)-bundles over the Wolf spaces whose total space is non-formal. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.08806v2-abstract-full').style.display = 'none'; document.getElementById('1709.08806v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 19 November, 2017; v1 submitted 25 September, 2017; originally announced September 2017. Comments: 31 pages, no figures. V2. Improved version, title changed MSC Class: 53C25; 55P62; 57N65; 55S30; 53C26 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1709.05724">arXiv:1709.05724</a>  [<a href="https://arxiv.org/pdf/1709.05724">pdf</a>, <a href="https://arxiv.org/format/1709.05724">other</a>]  <div class="tags is-inline-block"> math.AG math-ph math.AC math.CT </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1016/j.bulsci.2020.102871">10.1016/j.bulsci.2020.102871 </a> </div> </div> </div> A lax monoidal Topological Quantum Field Theory for representation varieties Authors: <a href="/search/math?searchtype=author&query=Gonz%C3%A1lez-Prieto%2C+%C3%81">脕ngel Gonz谩lez-Prieto</a>, <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on $G$. The construction is developed in a categorical-theoreti… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05724v2-abstract-full').style.display = 'inline'; document.getElementById('1709.05724v2-abstract-short').style.display = 'none';">▽ More</a> We construct a lax monoidal Topological Quantum Field Theory that computes Deligne-Hodge polynomials of representation varieties of the fundamental group of any closed manifold into any complex algebraic group $G$. As byproduct, we obtain formulas for these polynomials in terms of homomorphisms between the space of mixed Hodge modules on $G$. The construction is developed in a categorical-theoretic framework allowing its application to other situations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.05724v2-abstract-full').style.display = 'none'; document.getElementById('1709.05724v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 5 November, 2018; v1 submitted 17 September, 2017; originally announced September 2017. Comments: 26 pages, 4 figures. Added references. Added Section 5 with examples MSC Class: 57R56 (Primary); 14C30; 14D07; 14D21 (Secondary) Journal ref: Bulletin des Sciences Math茅matiques, Vol. 161, 2020, 102871 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1707.02998">arXiv:1707.02998</a>  [<a href="https://arxiv.org/pdf/1707.02998">pdf</a>, <a href="https://arxiv.org/ps/1707.02998">ps</a>, <a href="https://arxiv.org/format/1707.02998">other</a>]  <div class="tags is-inline-block"> math.DG hep-th </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.4310/PAMQ.2017.v13.n3.a4">10.4310/PAMQ.2017.v13.n3.a4 </a> </div> </div> </div> Orientability of the moduli space of Spin(7)-instantons Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Shahbazi%2C+C+S">C. S. Shahbazi</a> Abstract: Let $(M,惟)$ be a closed $8$-dimensional manifold equipped with a generically non-integrable $\mathrm{Spin}(7)$-structure $惟$. We prove that if $\mathrm{Hom}(H^{3}(M,\mathbb{Z}), \mathbb{Z}_{2}) = 0$ then the moduli space of irreducible $\mathrm{Spin}(7)$-instantons on $(M,惟)$ with gauge group $\mathrm{SU}(r)$, $r\geq 2$, is orientable. Let $(M,惟)$ be a closed $8$-dimensional manifold equipped with a generically non-integrable $\mathrm{Spin}(7)$-structure $惟$. We prove that if $\mathrm{Hom}(H^{3}(M,\mathbb{Z}), \mathbb{Z}_{2}) = 0$ then the moduli space of irreducible $\mathrm{Spin}(7)$-instantons on $(M,惟)$ with gauge group $\mathrm{SU}(r)$, $r\geq 2$, is orientable. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1707.02998v2-abstract-full').style.display = 'none'; document.getElementById('1707.02998v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 7 August, 2017; v1 submitted 10 July, 2017; originally announced July 2017. Comments: To appear in Pure and Applied Mathematics Quarterly Journal ref: Pure and Applied Mathematics Quarterly Volume 13 (2017) Number 3 Special Issue in Honor of Simon Donaldson </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1611.04127">arXiv:1611.04127</a>  [<a href="https://arxiv.org/pdf/1611.04127">pdf</a>, <a href="https://arxiv.org/ps/1611.04127">ps</a>, <a href="https://arxiv.org/format/1611.04127">other</a>]  <div class="tags is-inline-block"> math.DG hep-th </div> </div> Transversality for the moduli space of Spin(7)-instantons Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Shahbazi%2C+C+S">C. S. Shahbazi</a> Abstract: We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus. We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.04127v3-abstract-full').style.display = 'none'; document.getElementById('1611.04127v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 31 March, 2018; v1 submitted 13 November, 2016; originally announced November 2016. Comments: 34 pages, no figures. Several typos fixed and references added, title changed MSC Class: 53C38; 53C07; 53C25; 58D27 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1605.03024">arXiv:1605.03024</a>  [<a href="https://arxiv.org/pdf/1605.03024">pdf</a>, <a href="https://arxiv.org/ps/1605.03024">ps</a>, <a href="https://arxiv.org/format/1605.03024">other</a>]  <div class="tags is-inline-block"> math.DG math.AT math.SG </div> </div> Homotopic properties of K盲hler orbifolds Authors: <a href="/search/math?searchtype=author&query=Bazzoni%2C+G">Giovanni Bazzoni</a>, <a href="/search/math?searchtype=author&query=Biswas%2C+I">Indranil Biswas</a>, <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Tralle%2C+A">Aleksy Tralle</a> Abstract: We prove the formality and the evenness of odd-degree Betti numbers for compact K盲hler orbifolds, by adapting the classical proofs for K盲hler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any K盲hler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sas… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.03024v2-abstract-full').style.display = 'inline'; document.getElementById('1605.03024v2-abstract-short').style.display = 'none';">▽ More</a> We prove the formality and the evenness of odd-degree Betti numbers for compact K盲hler orbifolds, by adapting the classical proofs for K盲hler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any K盲hler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.03024v2-abstract-full').style.display = 'none'; document.getElementById('1605.03024v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 29 December, 2016; v1 submitted 10 May, 2016; originally announced May 2016. Comments: 29 pages, no figures. Background material on orbifolds substantially improved. Comments are welcome! arXiv admin note: text overlap with arXiv:1402.6861 MSC Class: 57R18; 55S30 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1601.06136">arXiv:1601.06136</a>  [<a href="https://arxiv.org/pdf/1601.06136">pdf</a>, <a href="https://arxiv.org/ps/1601.06136">ps</a>, <a href="https://arxiv.org/format/1601.06136">other</a>]  <div class="tags is-inline-block"> math.SG math.DG </div> </div> Homology Smale-Barden manifolds with K-contact and Sasakian structures Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Rojo%2C+J+A">Juan Angel Rojo</a>, <a href="/search/math?searchtype=author&query=Tralle%2C+A">Aleksy Tralle</a> Abstract: Koll谩r has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with $H_1(M,Z)=0$ which is K-contact but which carries no semi-regular Sasakian structures. Koll谩r has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with $H_1(M,Z)=0$ which is K-contact but which carries no semi-regular Sasakian structures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.06136v3-abstract-full').style.display = 'none'; document.getElementById('1601.06136v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 5 March, 2020; v1 submitted 22 January, 2016; originally announced January 2016. Comments: 31 pages, no figures; v2. small correction in the proof of Theorem 32 MSC Class: 53C25; 53D35; 57R17; 14J25 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1511.08930">arXiv:1511.08930</a>  [<a href="https://arxiv.org/pdf/1511.08930">pdf</a>, <a href="https://arxiv.org/ps/1511.08930">ps</a>, <a href="https://arxiv.org/format/1511.08930">other</a>]  <div class="tags is-inline-block"> math.DG </div> </div> Formality of 7-dimensional 3-Sasakian manifolds Authors: <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=Ivanov%2C+S">Stefan Ivanov</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not admit any 3-Sasakian structure. Examples of 7-dimensional simply connected compact… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.08930v1-abstract-full').style.display = 'inline'; document.getElementById('1511.08930v1-abstract-short').style.display = 'none';">▽ More</a> We prove that any simply connected compact 3-Sasakian manifold, of dimension seven, is formal if and only if its second Betti number is $b_2<2$. In the opposite, we show an example of a 7-dimensional Sasaki-Einstein manifold, with second Betti number $b_2\geq 2$, which is formal. Therefore, such an example does not admit any 3-Sasakian structure. Examples of 7-dimensional simply connected compact formal Sasakian manifolds, with $b_2\geq 2$, are also given. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.08930v1-abstract-full').style.display = 'none'; document.getElementById('1511.08930v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 28 November, 2015; originally announced November 2015. Comments: 10 pages MSC Class: 53C25; 55S30; 55P62 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1505.04451">arXiv:1505.04451</a>  [<a href="https://arxiv.org/pdf/1505.04451">pdf</a>, <a href="https://arxiv.org/ps/1505.04451">ps</a>, <a href="https://arxiv.org/format/1505.04451">other</a>]  <div class="tags is-inline-block"> math.GT math.AG </div> </div> The SL(3,C)-character variety of the figure eight knot Authors: <a href="/search/math?searchtype=author&query=Heusener%2C+M">Michael Heusener</a>, <a href="/search/math?searchtype=author&query=Munoz%2C+V">Vicente Munoz</a>, <a href="/search/math?searchtype=author&query=Porti%2C+J">Joan Porti</a> Abstract: We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.04451v1-abstract-full').style.display = 'inline'; document.getElementById('1505.04451v1-abstract-short').style.display = 'none';">▽ More</a> We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $Sym^2:SL(2,C) \to SL(3,C)$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1505.04451v1-abstract-full').style.display = 'none'; document.getElementById('1505.04451v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 17 May, 2015; originally announced May 2015. Comments: 30 pages, no figures. See http://mat.uab.cat/~porti/fig8.html for Mathematica notebooks and Sage worksheets associated to the computations in the paper MSC Class: 14D20; 57M25; 57M27 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1504.02451">arXiv:1504.02451</a>  [<a href="https://arxiv.org/pdf/1504.02451">pdf</a>, <a href="https://arxiv.org/ps/1504.02451">ps</a>, <a href="https://arxiv.org/format/1504.02451">other</a>]  <div class="tags is-inline-block"> math.SG math.AT math.DG </div> </div> Formality and the Lefschetz property in symplectic and cosymplectic geometry Authors: <a href="/search/math?searchtype=author&query=Bazzoni%2C+G">Giovanni Bazzoni</a>, <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We review topological properties of K盲hler and symplectic manifolds, and of their odd-dimensional counterparts, coK盲hler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the K盲hler/symplectic situation) and the $b_1=1$ case (in the coK盲hler/cosymplectic situation). We review topological properties of K盲hler and symplectic manifolds, and of their odd-dimensional counterparts, coK盲hler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the K盲hler/symplectic situation) and the $b_1=1$ case (in the coK盲hler/cosymplectic situation). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1504.02451v1-abstract-full').style.display = 'none'; document.getElementById('1504.02451v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 9 April, 2015; originally announced April 2015. Comments: 27 pages, no figures. Comments are welcome! MSC Class: 53C15; 55S30; 53D35; 55P62; 57R17 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1410.6045">arXiv:1410.6045</a>  [<a href="https://arxiv.org/pdf/1410.6045">pdf</a>, <a href="https://arxiv.org/ps/1410.6045">ps</a>, <a href="https://arxiv.org/format/1410.6045">other</a>]  <div class="tags is-inline-block"> math.SG math.AG math.DG </div> </div> A $6$-dimensional simply connected complex and symplectic manifold with no K盲hler metric Authors: <a href="/search/math?searchtype=author&query=Bazzoni%2C+G">Giovanni Bazzoni</a>, <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We construct a simply connected compact manifold which has complex and symplectic structures but does not admit K盲hler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form. We construct a simply connected compact manifold which has complex and symplectic structures but does not admit K盲hler metrics, in the lowest possible dimension where this can happen, that is, dimension 6. Such a manifold is automatically formal and has even odd-degree Betti numbers but it does not satisfy the Lefschetz property for any symplectic form. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1410.6045v2-abstract-full').style.display = 'none'; document.getElementById('1410.6045v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 14 November, 2014; v1 submitted 22 October, 2014; originally announced October 2014. Comments: 14 pages, no figures. Second version: minor changes in the introduction, exposition clarified and new references added. Comments are welcome! MSC Class: 53D05 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1409.4784">arXiv:1409.4784</a>  [<a href="https://arxiv.org/pdf/1409.4784">pdf</a>, <a href="https://arxiv.org/ps/1409.4784">ps</a>, <a href="https://arxiv.org/format/1409.4784">other</a>]  <div class="tags is-inline-block"> math.GT math.AG </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.2140/agt.2016.16.397">10.2140/agt.2016.16.397 </a> </div> </div> </div> Geometry of the SL(3,C)-character variety of torus knots Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Porti%2C+J">Joan Porti</a> Abstract: Let G be the fundamental group of the complement of the torus knot of type (m,n). This has a presentation G=<x,y|x^m=y^n>. We find the geometric description of the character variety X(G) of characters of representations of G into SL(3,C), GL(3,C) and PGL(3,C). Let G be the fundamental group of the complement of the torus knot of type (m,n). This has a presentation G=<x,y|x^m=y^n>. We find the geometric description of the character variety X(G) of characters of representations of G into SL(3,C), GL(3,C) and PGL(3,C). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.4784v2-abstract-full').style.display = 'none'; document.getElementById('1409.4784v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 8 March, 2020; v1 submitted 16 September, 2014; originally announced September 2014. Comments: 22 pages, no figures. v2. Corrected polynomial in theorem 8.3 MSC Class: 14D20; 57M25; 57M27 Journal ref: Algebr. Geom. Topol. 16 (2016) 397-426 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1408.2496">arXiv:1408.2496</a>  [<a href="https://arxiv.org/pdf/1408.2496">pdf</a>, <a href="https://arxiv.org/ps/1408.2496">ps</a>, <a href="https://arxiv.org/format/1408.2496">other</a>]  <div class="tags is-inline-block"> math.DG </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1007/s00209-015-1494-8">10.1007/s00209-015-1494-8 </a> </div> </div> </div> Simply-connected K-contact and Sasakian manifolds of dimension 7 Authors: <a href="/search/math?searchtype=author&query=Munoz%2C+V">Vicente Munoz</a>, <a href="/search/math?searchtype=author&query=Tralle%2C+A">Aleksy Tralle</a> Abstract: We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish. We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and that it is formal if and only if all its triple Massey products vanish. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1408.2496v2-abstract-full').style.display = 'none'; document.getElementById('1408.2496v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 18 August, 2014; v1 submitted 11 August, 2014; originally announced August 2014. Comments: 14 pages, some references added, several typos are corrected MSC Class: 53C25; 53D35; 57R17; 55P62 Journal ref: Mathematische Zeitschrift 281(2015), 457-470 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1407.6975">arXiv:1407.6975</a>  [<a href="https://arxiv.org/pdf/1407.6975">pdf</a>, <a href="https://arxiv.org/format/1407.6975">other</a>]  <div class="tags is-inline-block"> math.AG </div> </div> E-polynomials of the SL(2,C)-character varieties of surface groups Authors: <a href="/search/math?searchtype=author&query=Martinez-Martinez%2C+J">Javier Martinez-Martinez</a>, <a href="/search/math?searchtype=author&query=Munoz%2C+V">Vicente Munoz</a> Abstract: We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2,C), for any possible holonomy around the puncture. We follow the geometric technique introduced in arXiv:1106.6011, based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations. We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a once-punctured surface of any genus into SL(2,C), for any possible holonomy around the puncture. We follow the geometric technique introduced in arXiv:1106.6011, based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1407.6975v3-abstract-full').style.display = 'none'; document.getElementById('1407.6975v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 9 February, 2020; v1 submitted 25 July, 2014; originally announced July 2014. Comments: 22 pages, no figures. v2. References added. v3. Typo in matrix of page 17 corrected. arXiv admin note: text overlap with arXiv:1405.7120 MSC Class: 14C30; 14D20; 14L24; 32J25 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1405.7120">arXiv:1405.7120</a>  [<a href="https://arxiv.org/pdf/1405.7120">pdf</a>, <a href="https://arxiv.org/format/1405.7120">other</a>]  <div class="tags is-inline-block"> math.AG </div> </div> E-polynomial of SL(2,C)-character varieties of complex curves of genus 3 Authors: <a href="/search/math?searchtype=author&query=Martinez%2C+J">Javier Martinez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g=3 into $ SL(2,C), and also of the moduli space of twisted representations. The case of genus g=1,2 has already been done in [http://arxiv.org/abs/1106.6011]. We follow the geometric technique introduced in [http://arxiv.org/abs/1106.6011], based on stratifying the space of re… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.7120v2-abstract-full').style.display = 'inline'; document.getElementById('1405.7120v2-abstract-short').style.display = 'none';">▽ More</a> We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus g=3 into $ SL(2,C), and also of the moduli space of twisted representations. The case of genus g=1,2 has already been done in [http://arxiv.org/abs/1106.6011]. We follow the geometric technique introduced in [http://arxiv.org/abs/1106.6011], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.7120v2-abstract-full').style.display = 'none'; document.getElementById('1405.7120v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 28 August, 2014; v1 submitted 28 May, 2014; originally announced May 2014. Comments: 27 pages, 1 figure, References added MSC Class: 14C30; 14D20; 14L24; 32J25 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1405.0816">arXiv:1405.0816</a>  [<a href="https://arxiv.org/pdf/1405.0816">pdf</a>, <a href="https://arxiv.org/ps/1405.0816">ps</a>, <a href="https://arxiv.org/format/1405.0816">other</a>]  <div class="tags is-inline-block"> math.AG </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.2140/pjm.2016.282.173">10.2140/pjm.2016.282.173 </a> </div> </div> </div> E-polynomial of the SL(3,C)-character variety of free groups Authors: <a href="/search/math?searchtype=author&query=Lawton%2C+S">Sean Lawton</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3, C). Expanding upon existing techniques, we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler character… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.0816v2-abstract-full').style.display = 'inline'; document.getElementById('1405.0816v2-abstract-short').style.display = 'none';">▽ More</a> We compute the E-polynomial of the character variety of representations of a rank r free group in SL(3, C). Expanding upon existing techniques, we stratify the space of representations and compute the E-polynomial of each geometrically described stratum using fibrations. Consequently, we also determine the E-polynomial of its smooth, singular, and abelian loci and the corresponding Euler characteristic in each case. Along the way, we give new proofs of the SL(2,C), PGL(2,C), and PGL(3,C) cases. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1405.0816v2-abstract-full').style.display = 'none'; document.getElementById('1405.0816v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 12 September, 2015; v1 submitted 5 May, 2014; originally announced May 2014. Comments: 21 pages; accepted for publication at the Pacific Journal of Mathematics MSC Class: 14D20; 20C15; 14L30; 20E05 Journal ref: Pacific Journal of Mathematics, Vol. 282 (2016), No. 1, 173-202 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1404.7662">arXiv:1404.7662</a>  [<a href="https://arxiv.org/pdf/1404.7662">pdf</a>, <a href="https://arxiv.org/format/1404.7662">other</a>]  <div class="tags is-inline-block"> math.SG math.AT </div> </div> Manifolds which are complex and symplectic but not K盲hler Authors: <a href="/search/math?searchtype=author&query=Bazzoni%2C+G">Giovanni Bazzoni</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: The first example of a compact manifold admitting both complex and symplectic structures but not admitting a K盲hler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply-connected $8$-dimensional compact manifold of [\textsc{M. Fern谩ndez and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1404.7662v1-abstract-full').style.display = 'inline'; document.getElementById('1404.7662v1-abstract-short').style.display = 'none';">▽ More</a> The first example of a compact manifold admitting both complex and symplectic structures but not admitting a K盲hler structure is the renowned Kodaira-Thurston manifold. We review its construction and show that this paradigm is very general and is not related to the fundamental group. More specifically, we prove that the simply-connected $8$-dimensional compact manifold of [\textsc{M. Fern谩ndez and V. Mu帽oz}, \emph{An 8-dimensional non-formal simply connected symplectic manifold}, Ann. of Math. (2) \textbf{167}, no. 3, 1045--1054, 2008.] admits both symplectic and complex structures but does not carry K盲hler metrics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1404.7662v1-abstract-full').style.display = 'none'; document.getElementById('1404.7662v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 30 April, 2014; originally announced April 2014. Comments: 15 pages, 2 figures; comments are welcome! MSC Class: 53D05; 55P62 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1402.6861">arXiv:1402.6861</a>  [<a href="https://arxiv.org/pdf/1402.6861">pdf</a>, <a href="https://arxiv.org/ps/1402.6861">ps</a>, <a href="https://arxiv.org/format/1402.6861">other</a>]  <div class="tags is-inline-block"> math.DG math.AT </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> doi <a class="" href="https://doi.org/10.1112/jtopol/jtv044">10.1112/jtopol/jtv044 </a> </div> </div> </div> On formality of Sasakian manifolds Authors: <a href="/search/math?searchtype=author&query=Biswas%2C+I">Indranil Biswas</a>, <a href="/search/math?searchtype=author&query=Fern%C3%A1ndez%2C+M">Marisa Fern谩ndez</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Tralle%2C+A">Aleksy Tralle</a> Abstract: We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian m… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.6861v3-abstract-full').style.display = 'inline'; document.getElementById('1402.6861v3-abstract-short').style.display = 'none';">▽ More</a> We investigate some topological properties, in particular formality, of compact Sasakian manifolds. Answering some questions raised by Boyer and Galicki, we prove that all higher (than three) Massey products on any compact Sasakian manifold vanish. Hence, higher Massey products do obstruct Sasakian structures. Using this we produce a method of constructing simply connected K-contact non-Sasakian manifolds. On the other hand, for every $n \geq 3$, we exhibit the first examples of simply connected compact Sasakian manifolds of dimension $2n + 1$ which are non-formal. They are non-formal because they have a non-zero triple Massey product. We also prove that arithmetic lattices in some simple Lie groups cannot be the fundamental group of a compact Sasakian manifold. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1402.6861v3-abstract-full').style.display = 'none'; document.getElementById('1402.6861v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 11 November, 2015; v1 submitted 27 February, 2014; originally announced February 2014. Comments: 22 pages, no figures; v2. some corrections; v3. Accepted in J. Topology MSC Class: 57R18; 53C25; 55S30 Journal ref: Journal of Topology, 9 (2016) 161-180 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1311.4914">arXiv:1311.4914</a>  [<a href="https://arxiv.org/pdf/1311.4914">pdf</a>, <a href="https://arxiv.org/ps/1311.4914">ps</a>, <a href="https://arxiv.org/format/1311.4914">other</a>]  <div class="tags is-inline-block"> math.AG </div> </div> Hodge polynomials of the SL(2,C)-character variety of an elliptic curve with two marked points Authors: <a href="/search/math?searchtype=author&query=Logares%2C+M">Marina Logares</a>, <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2,C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recov… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1311.4914v3-abstract-full').style.display = 'inline'; document.getElementById('1311.4914v3-abstract-short').style.display = 'none';">▽ More</a> We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2,C). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1311.4914v3-abstract-full').style.display = 'none'; document.getElementById('1311.4914v3-abstract-short').style.display = 'inline';">△ Less</a> Submitted 9 February, 2020; v1 submitted 19 November, 2013; originally announced November 2013. Comments: 21 pages; v2. Accepted for publication in Internat. J. Math. v3. Mistake in polynomial of Theorem 1.1(3) corrected MSC Class: 14D20; 14C30; 14L24 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1309.1449">arXiv:1309.1449</a>  [<a href="https://arxiv.org/pdf/1309.1449">pdf</a>, <a href="https://arxiv.org/ps/1309.1449">ps</a>, <a href="https://arxiv.org/format/1309.1449">other</a>]  <div class="tags is-inline-block"> math.NT math.CV </div> </div> Unified treatment of Explicit and Trace Formulas via Poisson-Newton formula Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=P%C3%A9rez-Marco%2C+R">Ricardo P茅rez-Marco</a> Abstract: We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, explicit formulas in number theory and Selberg trace formulas in Riemannia… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.1449v2-abstract-full').style.display = 'inline'; document.getElementById('1309.1449v2-abstract-short').style.display = 'none';">▽ More</a> We prove that a Poisson-Newton formula, in a broad sense, is associated to each Dirichlet series with a meromorphic extension to the whole complex plane of finite order. These formulas simultaneously generalize the classical Poisson formula and Newton formulas for Newton sums. Classical Poisson formulas in Fourier analysis, explicit formulas in number theory and Selberg trace formulas in Riemannian geometry appear as special cases of our general Poisson-Newton formula. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1309.1449v2-abstract-full').style.display = 'none'; document.getElementById('1309.1449v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 27 October, 2014; v1 submitted 5 September, 2013; originally announced September 2013. Comments: 35 pages. This material has been extracted from the manuscript arxiv.org/abs/1301.6511 v2. Accepted in Comm. Math. Physics. arXiv admin note: substantial text overlap with arXiv:1301.6511 MSC Class: 11M06; 30B50; 11M36 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1307.8000">arXiv:1307.8000</a>  [<a href="https://arxiv.org/pdf/1307.8000">pdf</a>, <a href="https://arxiv.org/ps/1307.8000">ps</a>, <a href="https://arxiv.org/format/1307.8000">other</a>]  <div class="tags is-inline-block"> math.DG </div> </div> On rotation of complex structures Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a> Abstract: We put in a general framework the situations in which a Riemannian manifold admits a family of compatible complex structures, including hyperkahler metrics and the Spin-rotations of arxiv:1302.2846. We determine the (polystable) holomorphic bundles which are rotable, i.e., they remain holomorphic when we change a complex structure by a different one in the family. We put in a general framework the situations in which a Riemannian manifold admits a family of compatible complex structures, including hyperkahler metrics and the Spin-rotations of arxiv:1302.2846. We determine the (polystable) holomorphic bundles which are rotable, i.e., they remain holomorphic when we change a complex structure by a different one in the family. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1307.8000v2-abstract-full').style.display = 'none'; document.getElementById('1307.8000v2-abstract-short').style.display = 'inline';">△ Less</a> Submitted 9 January, 2014; v1 submitted 30 July, 2013; originally announced July 2013. Comments: 14 pages. Accepted in J. Geom. Phys MSC Class: 53C07; 53C26 </li> <li class="arxiv-result"> <div class="is-marginless"> <a href="https://arxiv.org/abs/1306.2165">arXiv:1306.2165</a>  [<a href="https://arxiv.org/pdf/1306.2165">pdf</a>, <a href="https://arxiv.org/ps/1306.2165">ps</a>, <a href="https://arxiv.org/format/1306.2165">other</a>]  <div class="tags is-inline-block"> math.CV </div> </div> On the genus of meromorphic functions Authors: <a href="/search/math?searchtype=author&query=Mu%C3%B1oz%2C+V">Vicente Mu帽oz</a>, <a href="/search/math?searchtype=author&query=Marco%2C+R+P">Ricardo P茅rez Marco</a> Abstract: We define the class of Left Located Divisor (LLD) meromorphic functions and their vertical order $m_0(f)$ and their convergence exponent $d(f)$. When $m_0(f)\leq d(f)$ we prove that their Weierstrass genus is minimal. This explains the phenomena that many classical functions have minimal Weierstrass genus, for example Dirichlet series, the $螕$-function, and trigonometric functions. We define the class of Left Located Divisor (LLD) meromorphic functions and their vertical order $m_0(f)$ and their convergence exponent $d(f)$. When $m_0(f)\leq d(f)$ we prove that their Weierstrass genus is minimal. This explains the phenomena that many classical functions have minimal Weierstrass genus, for example Dirichlet series, the $螕$-function, and trigonometric functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1306.2165v1-abstract-full').style.display = 'none'; document.getElementById('1306.2165v1-abstract-short').style.display = 'inline';">△ Less</a> Submitted 10 June, 2013; originally announced June 2013. Comments: 14 pages MSC Class: 30D30; 30B50; 30D15 </li> </ol> <nav class="pagination is-small is-centered breathe-horizontal" role="navigation" aria-label="pagination"> <a href="" class="pagination-previous is-invisible">Previous </a> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=50" class="pagination-next" >Next </a> <ul class="pagination-list"> <li> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=0" class="pagination-link is-current" aria-label="Goto page 1">1 </a> </li> <li> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=50" class="pagination-link " aria-label="Page 2" aria-current="page">2 </a> </li> <li> <a href="/search/?searchtype=author&query=Mu%C3%B1oz%2C+V&start=100" class="pagination-link " aria-label="Page 3" aria-current="page">3 </a> </li> </ul> </nav> <div class="is-hidden-tablet">  <a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>   </div> </div> </main> <footer> <div 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