MUFr in nLab

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title> MUFr in nLab </title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8" /> <meta name="robots" content="noindex,nofollow" /> <style type="text/css"> h1#pageName, div.info, .newWikiWord a, a.existingWikiWord, .newWikiWord a:hover, [actiontype="toggle"]:hover, #TextileHelp h3 { color: #226622; } </style> <link href="/stylesheets/instiki.css?1676280126" media="all" rel="stylesheet" type="text/css" /> <style type="text/css"><![CDATA[/*></style> <script src="/javascripts/prototype.js?1660229990" type="text/javascript"></script> <script src="/javascripts/effects.js?1660229990" type="text/javascript"></script> <script src="/javascripts/dragdrop.js?1660229990" type="text/javascript"></script> <script src="/javascripts/controls.js?1660229990" type="text/javascript"></script> <script src="/javascripts/application.js?1660229990" type="text/javascript"></script> <script type="text/javascript"> <![CDATA[//><!]]> </script> </head> <body> <div id="Container"> <textarea id='content' readonly=' readonly' rows='24' cols='60' > +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Cobordism theory +--{: .hide} [[!include cobordism theory -- contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea In joint generalization of the [[cobordism cohomology theories]] [[MU]] and [[MFr]] of [[closed manifold|closed]] $U$-manifolds and of $Fr$-manifolds, respectively, a _$(U,fr)$-manifold_ ([Conner-Floyd 66, Section 16](#ConnerFloyd66), [Conner-Smith 69, Sections 6, 13](#ConnerSmith69)) is a [[compact topological space|compact]] [[manifold with boundary]] equipped with [[unitary group]]-[[tangential structure]] on its [[stable tangent bundle]] and equipped with a [[trivial vector bundle|trivialization]] (stable [[framed manifold|framing]]) of that over the [[boundary]]. The corresponding [[bordism classes]] form a [[bordism ring]] denoted $\Omega^{\mathrm{U},fr}_\bullet$. ## Properties ### Representing spectrum {#RepresentingSpectrum} In generalization to how the complex [[cobordism ring]] $\Omega^U_{2k}$ is represented by [[homotopy classes]] of [[maps]] into the [[Thom spectrum]] [[MU]], so $\Omega^{\mathrm{U},fr}_{2k}$ is represented by maps into the [[quotient spaces]] $MU_{2k}/S^{2k}$ (for $S^{2k} = Th(\mathbb{C}^{k}) \to Th( \mathbb{C}^k \times_{\mathrm{U}(k)} E \mathrm{U}(k) ) = M \mathrm{U}_{2k}$ the canonical inclusion): \[ \label{InTermsOfHomotopyGroupsOfQuotientedThomSpace} \Omega^{(\mathrm{U},fr)}_\bullet \;=\; \pi_{\bullet + 2k} \big( MU_{2k}/S^{2k} \big) \,, \;\;\;\;\; \text{for any} \; 2k \geq \bullet + 2 \,. \] ([Conner-Floyd 66, p. 97](#ConnerFloyd66)) Hence the representing spectrum $M(\mathrm{U},fr)$ is the [[homotopy cofiber]] of the [[ring spectrum]] unit $1^{M\mathrm{U}} \;\colon\; \mathbb{S} \longrightarrow M \mathrm{U}$ out of the [[sphere spectrum]] ([Conner-Smith 69, p. 156 (41 of 106)](#ConnerSmith69), [Smith 71](#Smith71)) which deserves to be denoted $$ M(\mathrm{U},fr) \;\coloneqq\; M \mathrm{U} / \mathbb{S} \,, $$ but which in notation common around the [[Adams spectral sequence]] would be "$\Sigma \overline {M \mathrm{U}}$" (as in [Adams 74, theorem 15.1 page 319](Adams+spectral+sequence#Adams74)) or just "$\overline{ M \mathrm{U} }$" (e.g. [Hopkins 99, Cor. 5.3](Adams+spectral+sequence#Hopkins99)): \[ \label{AsUnitCofiber} \array{ \mathbb{S} & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & M \mathrm{U} \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast &\longrightarrow& M \mathrm{U}/ \mathbb{S} } \] So in terms of [[stable homotopy groups]] of this spectrum we have the $(\mathrm{U},fr)$-[[cobordism ring]] \[ \label{UFrCobordismRing} \Omega^{\mathrm{U},fr}_{\bullet} \;\coloneqq\; \pi_{\bullet} \big( M\mathrm{U}/\mathbb{S} \big) \] ### Boundary morphism to $MFr$ {#BoundaryMorphism} The realization (eq:AsUnitCofiber) makes it manifest (this is left implicit in [Conner-Floyd 66, p. 99](#ConnerFloyd66)) that there is a [[cohomology operation]] to [[MFr]] of the form \[ \label{BoundaryCohomologyOperation} \array{ M(\mathrm{U},fr) \;= & M \mathrm{U}/\mathbb{S} & \overset{ \;\;\; \partial \;\;\; }{\longrightarrow} & \Sigma \mathbb{S} & =\; \Sigma Mfr \\ \pi_{2d+2}\big( M(\mathrm{U},fr) \big) && \longrightarrow && \pi_{2d+1}\big( Mfr \big) } \,. \] Namely, $\partial$ is the second next step in the long [[homotopy cofiber]]-sequence starting with $1^{M \mathrm{U}}$. In terms of the [[pasting law]]: \[ \label{BoundaryOperationViaPastingLaw} \array{ \mathbb{S} & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & M \mathrm{U} & \longrightarrow & \ast \\ \big\downarrow & {}^{{}_{(po)}} & \big\downarrow & {}^{{}_{(po)}} & \big\downarrow \\ \ast & \longrightarrow & M \mathrm{U}/ \mathbb{S} & \underset{ \partial }{ \longrightarrow } & \Sigma \mathbb{S} } \] +-- {: .num_remark} ###### Remark The implicit idea in [Conner-Floyd 66, p. 99](#ConnerFloyd66) must be to see $\partial$ (eq:BoundaryCohomologyOperation) in terms of forming actual [[boundaries]] of representative [[manifolds with boundaries]] under a version of the [[Pontrjagin-Thom construction]]. This is certainly plausible, but not proven either, as they "forego the tedious details" on [p. 97](#ConnerFloyd66). NB: for the purpose on p. 99 they might just _define_ $\partial$ in this geometric way, but then the claim wrapping around p. 100-101 needs proof. This claim however is immediate from the abstract homotopy theory, namely it is just the continuation yet one step further along the long cofiber sequence -- this is Prop. \ref{AShortExactSequenceOfUFrBordismRings} below. =-- \linebreak ### Relation to $MU$ and $MFr$ {#RelationToMUAndFr} +-- {: .num_prop #AShortExactSequenceOfUFrBordismRings} ###### Proposition In [[positive number|positive]] degree, the underlying [[abelian groups]] of the [[bordism rings]] for [[MU]], [[MFr]] and $MUFr$ (eq:UFrCobordismRing) sit in [[short exact sequences]] of this form: \[ \label{ShortExactSequenceOfUFrBordismRings} 0 \to \Omega^{\mathrm{U}}_{2n + 2} \overset{i}{\longrightarrow} \Omega^{\mathrm{U},fr}_{2n + 2} \overset{\partial}{ \longrightarrow } \Omega^{fr}_{2n + 1} \to 0 \,, \phantom{AAAA} n \in \mathbb{N} \,, \] where $i$ is the evident inclusion, while $\partial$ is the [[boundary]] homomorphism from [above](#BoundaryMorphism). =-- This is stated without comment in [Conner-Floyd 66, p. 99](#ConnerFloyd66). The beginning of an argument appears inside the proof of [CF66, Thm. 16.2 (p. 100)](#ConnerFloyd66), attributed there to [[Peter Landweber]] (see Remark \ref{BoundaryOperationFromFrBordToUFrBordIsSurjective} below). The idea for how to complete the argument is a little more explicit in [Stong 68, p. 102](#Stong68). The following is the complete and quick proof using the formulation (eq:BoundaryOperationViaPastingLaw) via abstract homotopy [above](#BoundaryMorphism): +-- {: .proof} ###### Proof We have the [[long exact sequence of homotopy groups]] ([[long exact sequence in generalized cohomology]] on [[spheres]]) obtained from the [[cofiber sequence]] $\mathbb{S} \overset{1^{M\mathrm{U}}}{\longrightarrow} M \mathrm{U} \to M \mathrm{U}/\mathbb{S} \overset{\partial}{\to} \Sigma \mathbb{S}$ (eq:BoundaryOperationViaPastingLaw), the relevant part of which looks as follows: \[ \label{SToMULongExactSequenceOfHomotopyGroups} \array{ \overset{ \mathclap{ \color{darkblue} pure \; torsion } }{ \overbrace{ \pi_{2d+2} \big( \mathbb{S} \big) } } & \overset{ 1^{M\mathrm{U}} }{ \longrightarrow } & \overset{ \mathclap{ \color{darkblue} free \; abelian } }{ \overbrace{ \pi_{2d+2} \big( M\mathrm{U} \big) } } & \overset{ }{\longrightarrow} & \pi_{2d+2} \big( M\mathrm{U}/\mathbb{S} \big) & \overset{ \partial }{\longrightarrow} & \pi_{2d+1}\big(\mathbb{S}\big) &\longrightarrow& \overset{ \color{darkblue} trivial }{ \overbrace{ \pi_{2d+1}\big(M\mathrm{U}\big) } } \\ \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} && \big\downarrow{}^{\mathrlap{=}} \\ \Omega^{fr}_{2d+2} & \underset{ \color{green} 0 }{ \longrightarrow } & \Omega^{\mathrm{U}}_{2d+2} & \underset{ i }{\longrightarrow} & \Omega^{(\mathrm{U},fr)}_{2d+2} & \underset{ \partial }{\longrightarrow} & \Omega^{fr}_{2d + 1} & \underset{ \color{green} 0 }{\longrightarrow} & 0 } \] Observing now that the [[stable homotopy groups]] of $M\mathrm{U}$ are [[free abelian groups]] concentrated in [[even number|even]] degrees (by [this theorem](MU#RelationToCobordismRing) at _[[MU]]_) it follows that: 1. the rightmost morphism shown in (eq:SToMULongExactSequenceOfHomotopyGroups) is the [[zero morphism]] since its [[codomain]] is [[zero object|zero]]; 1. the leftmost morphism shown in (eq:SToMULongExactSequenceOfHomotopyGroups) is the [[zero morphism]], since the [[stable homotopy groups of spheres]] are all pure [[torsion groups]] in [[positive number|positive]] degrees (by the [[Serre finiteness theorem]]), and the only morphism from a torsion group to a [[free abelian group]] is the zero morphism. =-- +-- {: .num_remark #BoundaryOperationFromFrBordToUFrBordIsSurjective} ###### Remark Here is an explicit construction of a [[lift]] through the boundary map in (eq:ShortExactSequenceOfUFrBordismRings), depending on a choice of trivialization in $\pi_{2d-1}\big( M\mathrm{U}\big)$: (The following is essentially a streamlined account of the construction in the first half of the proof of [Conner-Floyd 66, Thm. 16.2](#ConnerFloyd66); for more and a more abstract perspective see at _[[d-invariant]]_ the section _[Trivializations of the d-invariant](d-invariant#TrivializationsOfThedInvariant)_): Let $$ \big[ \Sigma^\infty S^{2(n+d)-1} \overset{ \Sigma^{\infty} {\color{green} c } }{\longrightarrow} \Sigma^\infty S^{2n} \big] \;\in\; \pi^s_{2d-1} \;\simeq\; \Omega^{fr}_{2d-1} $$ be a given class in [[stable Cohomotopy]], hence in the [[MFr]]-[[cobordism ring]], under the [[Pontryagin-Thom isomorphism|PT isomorphism]]. (We write this as the [[stabilization]] of a class ${\color{green} c}$ in unstable [[Cohomotopy]] just for emphasis that we can.) Consider then following [[homotopy coherent diagram|homotopy]] [[pasting diagram]]: \begin{imagefromfile} "file_name": "LiftingFromUBordismToUFrBordism.jpg", "width": 600, "unit": "px", "margin": { "top": -20, "bottom": 20, "right": 0, "left": 10 }, "caption": "From [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)" \end{imagefromfile} Here all squares are [[homotopy pushout]]-squares, they arise as follows (beware that we say "square" for any _single_ cell and "rectangle" for the pasting composite of any adjacent _pair_ of cells): * The top square witnesses the [[homotopy cofiber]] $C_c$, defined thereby; * the left rectangle is a homotopy pushout by definition of [[suspension]], * hence the bottom left square is so by the [[pasting law]]. * Again via the first point, a dashed morphism exists as shown, witnessing the fact that the pullback of the class $\Sigma^{2n}(1^{M\mathrm{U}})$ to an odd-dimensional square vanishes (as does every $MU$-class); * with this, the bottom left rectangle exists, and it is a homotopy pushout by definition of $M \mathrm{U}/\mathbb{S}$ (eq:BoundaryOperationViaPastingLaw); * hence the morphism $\color{magenta} M^{2 d}$ exists, and the resulting bottom middle square is a homotopy pushout, again by the [[pasting law]]. * Now the bottom right rectangle is defined to be a homotopy pushout, and thus looks as shown by the [[pasting law]]; * therefore the bottom right square exists, and it is a homotopy pushout by the [[pasting law]]. But now the commuting three morphism in the very bottom and right part of the diagram shows that ${\color{magenta}M^{2d}} \in \pi_{2d}\big( M(U,fr) \big)$ is a lift of ${\color{green} c} \in \pi_{2d-1}(M Fr)$ through $\partial$. =-- ### Relation to Todd classes and the e-invariant +-- {: .num_prop #EInvariantIsToddClassOnCoboundingUFrManifold} ###### Proposition **([[e-invariant is Todd class of cobounding (U,fr)-manifold]])** Evaluation of the [[Todd class]] on $(U,fr)$-manifolds yields [[rational numbers]] which are [[integers]] on actual $U$-manifolds. It follows with the [[short exact sequence]] (eq:ShortExactSequenceOfUFrBordismRings) that assigning to $Fr$-manifolds the Todd class of any of their cobounding $(U,fr)$-manifolds yields a well-defined element in [[Q/Z]]. Under the [[Pontrjagin-Thom isomorphism]] between the [[framed bordism ring]] and the [[stable homotopy group of spheres]] $\pi^s_\bullet$, this assignment coincides with the [[Adams e-invariant]] in [its Q/Z-incarnation](Adams+e-invariant#TheEInvariantAsAnElementOfQModZ): \[ \label{ToddClassesOnShortExactSequenceOfUFrBordismRings} \array{ 0 \to & \Omega^{\mathrm{U}}_{\bullet+1} & \overset{i}{\longrightarrow} & \Omega^{\mathrm{U},fr}_{\bullet+1} & \overset{\partial}{ \longrightarrow } & \Omega^{fr}_\bullet & \simeq & \pi^s_\bullet \\ & \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{\mathrlap{Td}} && \big\downarrow{}^{} && \big\downarrow{}^{e} \\ 0 \to & \mathbb{Z} &\overset{\;\;\;\;\;}{\hookrightarrow}& \mathbb{Q} &\overset{\;\;\;\;}{\longrightarrow}& \mathbb{Q}/\mathbb{Z} &=& \mathbb{Q}/\mathbb{Z} } \,, \] =-- ([Conner-Floyd 66, Theorem 16.2](#ConnerFloyd66)) The first step in the proof of (eq:ToddClassesOnShortExactSequenceOfUFrBordismRings) is the observation ([Conner-Floyd 66, p. 100-101](#ConnerFloyd66)) that the representing map (eq:InTermsOfHomotopyGroupsOfQuotientedThomSpace) for a $(U,fr)$-manifold $M^{2k}$ cobounding a $Fr$-manifold represented by a map $f$ is given by the following [[homotopy coherent diagram|homotopy]] [[pasting diagram]] (see also at _[[Hopf invariant]] -- [In generalized cohomology](Hopf+invariant#InGeneralizedCohomology)_): \begin{imagefromfile} "file_name": "CoboundingUFrManifoldDiagrammatically.jpg", "web": "nlab", "width": 470, "unit": "px", "margin": { "top": -20, "right": 0, "bottom": 20, "left": 20, "unit": "px" }, "alt": "homotopy pasting diagram exhibiting cobounding UFr-manifolds", "caption": "from [SS21](https://ncatlab.org/schreiber/show/Equivariant+Cohomotopy+and+Oriented+Cohomology+Theory)" \end{imagefromfile} ## Related concepts [[!include flavours of cobordism cohomology theories -- table]] ## References The concept of $(U,fr)$-bordism theory and its relation to the [[e-invariant]] originates with: * {#ConnerFloyd66} [[Pierre Conner]], [[Edwin Floyd]], Section 16 of: _[[The Relation of Cobordism to K-Theories]]_, Lecture Notes in Mathematics __28__ Springer 1966 ([doi:10.1007/BFb0071091](https://link.springer.com/book/10.1007/BFb0071091), [MR216511](http://www.ams.org/mathscinet-getitem?mr=216511)) * {#ConnerSmith69} [[Pierre Conner]], [[Larry Smith]], Section 6 of: _On the complex bordism of finite complexes_, Publications Math茅matiques de l'IH脡S, Tome 37 (1969) , pp. 117-221 ([numdam:PMIHES_1969__37__117_0](http://www.numdam.org/item/?id=PMIHES_1969__37__117_0)) Analogous discussion for [[MO]]-bordism with [[MSO]]-boundaries: * G. E. Mitchell, _Bordism of Manifolds with Oriented Boundaries_, Proceedings of the American Mathematical Society Vol. 47, No. 1 (Jan., 1975), pp. 208-214 ([doi:10.2307/2040234](https://doi.org/10.2307/2040234)) Analogous discussion for [[MOFr]] is in * {#Stong68} [[Robert Stong]], p. 102 of: _Notes on Cobordism theory_, Princeton University Press, 1968 ([toc pdf](http://pi.math.virginia.edu/StongConf/Stongbookcontents.pdf), [ISBN:9780691649016](http://press.princeton.edu/titles/6465.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/stongcob.pdf)) See also * {#Smith71} [[Larry Smith]], _On characteristic numbers of almost complex manifolds with framed boundaries_, Topology Volume 10, Issue 3, August 1971, Pages 237-256 (<a href="https://doi.org/10.1016/0040-9383(71)90008-5">doi:10.1016/0040-9383(71)90008-5</a>) Generalization to [[manifolds with corners]] and relation to the [[f-invariant]]: * Gerd Laures, _On cobordism of manifolds with corners_, Trans. Amer. Math. Soc. 352 (2000) ([doi:10.1090/S0002-9947-00-02676-3](https://doi.org/10.1090/S0002-9947-00-02676-3)) [[!redirects (U,fr)-manifold]] [[!redirects (U,fr)-manifolds]] [[!redirects (U,fr)-bordism]] [[!redirects (U,fr)-bordisms]] [[!redirects (U,fr)-bordism ring]] [[!redirects (U,fr)-bordism rings]] [[!redirects (U,fr)-cobordism]] [[!redirects (U,fr)-cobordisms]] [[!redirects (U,fr)-cobordism ring]] [[!redirects (U,fr)-cobordism rings]] </textarea> </div>  </body> </html>

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MUFr in nLab