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class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> <h4 id="homotopy_theory">Homotopy theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#details'>Details</a></li> <ul> <li><a href='#maps_from_a_riemann_surface_to_a_projective_space'>Maps from a Riemann surface to a projective space</a></li> <li><a href='#MapsBetweenProjectiveSpaces'>Maps between projective spaces</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#maps_out_of_riemann_surfaces'>Maps out of Riemann surfaces</a></li> <li><a href='#maps_between_projective_spaces_2'>Maps between projective spaces</a></li> <li><a href='#maps_from_projective_spaces_to_toric_varieties'>Maps from projective spaces to toric varieties</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>On the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+type">stable</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> of (regular) <a class="existingWikiWord" href="/nlab/show/rational+maps">rational maps</a>.</p> <p>Under good conditions, the <a class="existingWikiWord" href="/nlab/show/topological+subspace">subspace inclusion</a> of the space of <a class="existingWikiWord" href="/nlab/show/rational+maps">rational maps</a> (regular, see Rem. <a class="maruku-ref" href="#NotionsOfRationalMaps"></a>), between given projective <a class="existingWikiWord" href="/nlab/show/complex+manifolds">complex manifolds</a> or <a class="existingWikiWord" href="/nlab/show/algebraic+varieties">algebraic varieties</a>, into the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> of all <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a><br />induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> in integral <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> in low degrees, while for maps out of the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a> this is even an isomorphism on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> in low degrees:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Maps</mi> <mi>rat</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>↪</mo><mfrac linethickness="0"><mrow><mrow><mi>homology</mi><mspace width="thickmathspace"></mspace><mi>iso</mi></mrow></mrow><mrow><mrow><mi>in</mi><mspace width="thickmathspace"></mspace><mi>low</mi><mspace width="thickmathspace"></mspace><mi>degree</mi></mrow></mrow></mfrac></mover><msub><mi>Maps</mi> <mi>cts</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Maps_{rat}(X_1, X_2) \xhookrightarrow{ {homology\; iso} \atop {in\; low\; degree} } Maps_{cts}(X_1, X_2) \,. </annotation></semantics></math></div> <p>This phenomenon originates in results of <a href="#Segal79">Segal 1979</a> and is commonly referred to by Segal’s name (e.g. “theorems of Segal-type” in <a href="#FriedlanderLawson97">Friedlander & Lawson 1997, Sec. 5.C</a>.</p> <p>Whenever this holds it provides</p> <ol> <li> <p>from left to right: <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theoretic</a> tools for analyzing <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of rational hypersurfaces;</p> </li> <li> <p>from right to left: small algebraic models for <a class="existingWikiWord" href="/nlab/show/stable+homotopy+types">stable</a> <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> of <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a></p> </li> </ol> <p>at least up to some dimension.</p> <h2 id="details">Details</h2> <p>Some remarks on the terminology being used:</p> <p> <div class='num_remark' id='NotionsOfDegree'> <h6>Remark</h6> <p><strong>(“degree”)</strong><br /> Most or all of the following statement invoke an <a class="existingWikiWord" href="/nlab/show/integer">integer</a> “degree” of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>. Beware that this is <em>not</em> the <a class="existingWikiWord" href="/nlab/show/degree+of+a+continuous+function">degree of a continuous function</a> (see there) in the usual sense of <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, except in special cases (such as the archetypical example <a class="maruku-ref" href="#TheArchetypicalExample"></a>).</p> </div> </p> <p> <div class='num_remark' id='NotionsOfRationalMaps'> <h6>Remark</h6> <p><strong>(“rational maps”)</strong> <br /> It it tradition (starting with Segal) to speak of <a class="existingWikiWord" href="/nlab/show/rational+maps">rational maps</a> in the following, but in the end the focus on <em>regular rational maps</em> (“morphisms”: e.g. <a href="#FriedlanderLawson97">Friedlander & Lawson 1997, p. 27</a>)), as is necessary to regard them as <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> defined everywhere on the given <a class="existingWikiWord" href="/nlab/show/domain">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math>.</p> <p>In many cases of interest, such as when the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math> is a non-singular <a class="existingWikiWord" href="/nlab/show/complex+curve">complex curve</a>/<a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> and the <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_2</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a>, then all rational maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_2</annotation></semantics></math> are automatically regular (e.g. <a href="rational+map#ShafarevichVol1">Shafarevich Vol1, Cor. 2.3</a>).</p> </div> For review of more details see <a href="#Havlicek92">Havlicek 92, §1</a>.</p> <h3 id="maps_from_a_riemann_surface_to_a_projective_space">Maps from a Riemann surface to a projective space</h3> <ul> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><msub><mi>ℕ</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}_+</annotation></semantics></math>, consider <a class="existingWikiWord" href="/nlab/show/complex+projective+n-space">complex projective n-space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^n</annotation></semantics></math>.</p> </li> <li> <p>Say that a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Σ</mi> <mn>2</mn></msub><mo>→</mo><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f \;\colon\; \Sigma_2 \to \mathbb{C}P^n</annotation></semantics></math> out of a 2-<a class="existingWikiWord" href="/nlab/show/dimension+of+a+manifold">dimensional</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> has <em>degree</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math> (Rem. <a class="maruku-ref" href="#NotionsOfDegree"></a>) if the <a class="existingWikiWord" href="/nlab/show/pullback+in+cohomology">pullback</a> of the generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>ℤ</mi><mo>≃</mo><msup><mi>H</mi> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">1 \in \mathbb{Z} \simeq H^2\big( \mathbb{C}P^n;\, \mathbb{Z}\big)</annotation></semantics></math> (see <a href="complex+projective+space#OrdinaryHomologyAndCohomology">here</a>) is</p> <div class="maruku-equation" id="eq:DegreeOfAMapToCPn"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>d</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>≃</mo><mspace width="thinmathspace"></mspace><msup><mi>H</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mn>2</mn></msub><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^\ast(1) \,=\, d \,\in\, \mathbb{Z} \,\simeq\, H^2(\Sigma_2;\, \mathbb{Z}) \,. </annotation></semantics></math></div></li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>Σ</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">g_\Sigma \in \mathbb{N}</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/genus+of+a+Riemann+surface">genus</a>.</p> </li> </ul> <p> <div class='num_prop' id='SegalOnHomotopyTypeOfRationalMapsIntoCPn'> <h6>Proposition</h6> <p><strong>(Segal’s theorem)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, the inclusion</p> <div class="maruku-equation" id="eq:RationalMappingSpaceInsideContinuousMappingSpace"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Maps</mi> <mi>rat</mi> <mrow><mi>deg</mi><mo>=</mo><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Σ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>Maps</mi> <mi>cts</mi> <mrow><mi>deg</mi><mo>=</mo><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Σ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> Maps_{ {rat} }^{deg = d} \big( \Sigma ,\, \mathbb{C}P^n \big) \xhookrightarrow{ \;\; i \;\; } Maps_{ {cts} }^{deg = d} \big( \Sigma ,\, \mathbb{C}P^n \big) </annotation></semantics></math></div> <p>of</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of <a class="existingWikiWord" href="/nlab/show/rational+maps">rational maps</a> to <a class="existingWikiWord" href="/nlab/show/complex+projective+n-space">complex projective n-space</a></p> <p>(the <em><a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of rational <a class="existingWikiWord" href="/nlab/show/complex+curves">complex curves</a> in <a class="existingWikiWord" href="/nlab/show/complex+projective+n-space"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℂ</mi> <msup><mi>P</mi> <mi>n</mi></msup> </mrow> <annotation encoding="application/x-tex">\mathbb{C}P^n</annotation> </semantics> </math></a></em>)</p> </li> </ul> <p>into</p> <ul> <li>the <a class="existingWikiWord" href="/nlab/show/connected+component">connected component</a> of maps of <a class="existingWikiWord" href="/nlab/show/degree+of+a+map">degree</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> (with the <a class="existingWikiWord" href="/nlab/show/compact+open+topology">compact open topology</a>) of all <a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a></li> </ul> <p>is</p> <ul> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>Σ</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">g_\Sigma = 0</annotation></semantics></math> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a>):</p> <p>a <a class="existingWikiWord" href="/nlab/show/n-equivalence"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>d</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">d (2 n -1)</annotation> </semantics> </math>-equivalence</a>, hence</p> <p>an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo><</mo><mi>d</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lt d (2 n -1)</annotation></semantics></math></p> <p>and a <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a> on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d (2 n - 1)</annotation></semantics></math></p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>Σ</mi></msub><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">g_\Sigma \geq 1</annotation></semantics></math>:</p> <p>an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary</a> <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a> in degrees <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo><</mo><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lt (d - 2g) (2 n -1)</annotation></semantics></math></p> <p>and a <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a> on <a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mn>2</mn><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d - 2g) (2 n -1)</annotation></semantics></math>.</p> </li> </ul> <p></p> </div> </p> <p>This is due to <a href="#Segal79">Segal 1979, Prop. 1.2, 1.3</a> (bewaring the Note on terminology on p. 44). The analogous statement for rational curves in <a class="existingWikiWord" href="/nlab/show/real+projective+spaces">real projective spaces</a> is in <a href="#Mostovoy01">Mostovoy 01</a>.</p> <p> <div class='num_remark' id='TheArchetypicalExample'> <h6>Example</h6> <p><strong>(the archetypical case)</strong> <br /> In the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>=</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\Sigma = S^2</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a> with its <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a>, so that both <a class="existingWikiWord" href="/nlab/show/domain">domain</a> and <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> are the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^1</annotation></semantics></math>, Prop. <a class="maruku-ref" href="#SegalOnHomotopyTypeOfRationalMapsIntoCPn"></a> says that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Maps</mi> <mi>rat</mi> <mrow><mi>deg</mi><mo>=</mo><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>Maps</mi> <mi>top</mi> <mrow><mi>deg</mi><mo>=</mo><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mn>2</mn></msup><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>S</mi> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> Maps_{ {rat} }^{deg = d} \big( \mathbb{C}P^1 ,\, \mathbb{C}P^1 \big) \xhookrightarrow{ \;\; i \;\; } Maps_{ {top} }^{deg = d} \big( S^2 ,\, S^2 \big) </annotation></semantics></math></div> <p>is an isomorphism on <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> up to degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">\leq d</annotation></semantics></math>.</p> </div> (<a href="#Segal79">Segal 1979, Prop. 1.1’</a>)</p> <p> <div class='num_remark' id='RelationToYangMillsMonopoles'> <h6>Remark</h6> <p><strong>(relation to <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopoles">Yang-Mills monopoles</a>)</strong> <br /> Example <a class="maruku-ref" href="#TheArchetypicalExample"></a> controls the classification of <a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopoles">Yang-Mills monopoles</a>. See there for more</p> </div> </p> <p> <div class='num_remark' id='RelationToGromovWittenTheory'> <h6>Remark</h6> <p><strong>(relation to <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a>)</strong> A compactification and <a class="existingWikiWord" href="/nlab/show/quotient+stack">quotient stack</a> of the space of rational maps in <a class="maruku-eqref" href="#eq:RationalMappingSpaceInsideContinuousMappingSpace">(2)</a> is considered in <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a>, e.g. <a href="Gromov-Witten+theory#Bertram02">Bertram 2002, p. 9</a>.</p> </div> </p> <p> <div class='num_remark' id='RelationToTwistorStringTheory'> <h6>Remark</h6> <p><strong>(relation to <a class="existingWikiWord" href="/nlab/show/twistor+string+theory">twistor string theory</a>)</strong> <br /> In the context of <a class="existingWikiWord" href="/nlab/show/twistor+string+theory">twistor string theory</a>, the spaces of rational maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>3</mn></msup></mrow><annotation encoding="application/x-tex">\Sigma \to \mathbb{C}P^3</annotation></semantics></math> <a class="maruku-eqref" href="#eq:RationalMappingSpaceInsideContinuousMappingSpace">(2)</a> are interpreted as <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of <a class="existingWikiWord" href="/nlab/show/D1-brane">D1-brane</a>-<a class="existingWikiWord" href="/nlab/show/instantons">instantons</a> in the <a class="existingWikiWord" href="/nlab/show/twistor+space">twistor space</a> <a class="existingWikiWord" href="/nlab/show/complex+projective+3-space"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℂ</mi> <msup><mi>P</mi> <mn>3</mn></msup> </mrow> <annotation encoding="application/x-tex">\mathbb{C}P^3</annotation> </semantics> </math></a> (<a href="twistor+string+theory#Witten04">Witten 2004, Sec. 3</a>).</p> <p>Such rational maps are also argued to encode <a class="existingWikiWord" href="/nlab/show/scattering+amplitudes">scattering amplitudes</a> in <a class="existingWikiWord" href="/nlab/show/D%3D4+N%3D8+supergravity">D=4 N=8 supergravity</a> (<a href="twistor+string+theory#CachazoSkinner12">Cachazo & Skinner 2012</a>, <a href="twistor+string+theory#Adamo15">Adamo 2015</a>).</p> <p>Here the number of poles in the rational function is the number <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> of particles in the <a class="existingWikiWord" href="/nlab/show/n-point+function">n-point function</a>, and the genus and degree encode the particle’s <a class="existingWikiWord" href="/nlab/show/helicity">helicity</a> and the <a class="existingWikiWord" href="/nlab/show/loop+order">loop order</a> of the <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a>.</p> </div> </p> <p> <div class='num_remark' id='ComaprisonToHomotopicalOkaPrinciple'> <h6>Remark</h6> <p><strong>(comparison to the <a class="existingWikiWord" href="/nlab/show/homotopical+Oka+principle">homotopical Oka principle</a>)</strong> <br /> Prop. <a class="maruku-ref" href="#SegalOnHomotopyTypeOfRationalMapsIntoCPn"></a> may be compared to the <a href="Oka+principle#HomotopicalOkaPrinciple">homotopical Oka principle</a>, which applies (since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^n</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/Oka+manifold">Oka manifold</a> by <a href="Oka+manifold#ComplexProjectiveSpaceIsOkaManifold">this Prop.</a>) to the complementary case of connected <em>non-compact</em> (“open”) Riemann surfaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> (which are <a class="existingWikiWord" href="/nlab/show/Stein+manifolds">Stein manifolds</a> by <a href="Stein+manifold#SteinSurfacesAreOpenRiemannSurfaces">this Example</a>), in which case it says that the corresponding inclusion of the space of <a class="existingWikiWord" href="/nlab/show/holomorphic+maps">holomorphic maps</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Maps</mi> <mi>hol</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Σ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msub><mi>Maps</mi> <mi>top</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Σ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> Maps_{ {hol} } \big( \Sigma ,\, \mathbb{C}P^n \big) \xhookrightarrow{ \;\; i \;\; } Maps_{ {top} } \big( \Sigma ,\, \mathbb{C}P^n \big) </annotation></semantics></math></div> <p>induces an isomorphism on all <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a>, hence is a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> – reflecting the fact that non-compactness of the Riemann surfaces and absence of any asymptotic boundary condition provides a large supply of holomorphic functions.</p> </div> </p> <p>In fact:</p> <p> <div class='num_prop' id='RationalMapsToCPnAsConfigurationsOfPoints'> <h6>Proposition</h6> <p>The full <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the space of <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">pointed</a> rational maps from the <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a> to <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^n</annotation></semantics></math> of algebraic degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> is that of the <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of at most <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>d</mi> </mrow> <annotation encoding="application/x-tex">d</annotation> </semantics> </math> points</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> with labels in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">S^{2n-1}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Maps</mi> <mi>rat</mi> <mrow><mi>deg</mi><mo>=</mo><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Σ</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><msub><mo>≃</mo> <mi>htpy</mi></msub><mspace width="thickmathspace"></mspace><munder><mi>Conf</mi><mrow><mo>≤</mo><mi>d</mi></mrow></munder><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>;</mo><msup><mi>S</mi> <mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> Maps_{ {rat} }^{deg = d} \big( \Sigma ,\, \mathbb{C}P^n \big) \;\simeq_{htpy}\; \underset{ \leq d}{Conf} \big( \mathbb{R}^2; S^{2k+1} \big) </annotation></semantics></math></div> <p></p> </div> </p> <p>(<a href="#CohenShimamoto91">Cohen & Shimamoto 91, Theorem 1</a>)</p> <h3 id="MapsBetweenProjectiveSpaces">Maps between projective spaces</h3> <p>Generalization to higher dimensional domain spaces:</p> <ul> <li> <p>Say that the <em>degree</em> of a <a class="existingWikiWord" href="/nlab/show/rational+map">rational map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f \;\colon\; \mathbb{C}P^{n_1} \to \mathbb{C}P^{n_2}</annotation></semantics></math> between two <a class="existingWikiWord" href="/nlab/show/complex+projective+spaces">complex projective spaces</a> is the degree of the <a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a> that define it.</p> </li> <li> <p>In generalization of <a class="maruku-eqref" href="#eq:DegreeOfAMapToCPn">(1)</a>, say that a <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f \;\colon\; \mathbb{C}P^{n_1} \to \mathbb{C}P^{n_2}</annotation></semantics></math> between two <a class="existingWikiWord" href="/nlab/show/complex+projective+spaces">complex projective spaces</a> has <em>degree</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math> (Rem. <a class="maruku-ref" href="#NotionsOfDegree"></a>) if this is the induced factor for <a class="existingWikiWord" href="/nlab/show/pullback+in+cohomology">pullback in</a> their second <a class="existingWikiWord" href="/nlab/show/integers">integral</a> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> (see <a href="complex+projective+space#OrdinaryHomologyAndCohomology">here</a>)</p> <div class="maruku-equation" id="eq:DegreeOfAMapBetweenComplexProjectiveSpaces"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo>≃</mo><msup><mi>H</mi> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover></mtd> <mtd><msup><mi>H</mi> <mn>2</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>≃</mo><mi>ℤ</mi></mtd></mtr> <mtr><mtd><mn>1</mn></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>d</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z} \simeq H^2\big( \mathbb{C}P^{n_2};\, \mathbb{Z}\big) &\xrightarrow{ f^\ast }& H^2\big( \mathbb{C}P^{n_1};\, \mathbb{Z}\big) \simeq \mathbb{Z} \\ 1 &\mapsto& d } \,. </annotation></semantics></math></div></li> </ul> <p> <div class='num_prop' id='HomologyEquivalenceForMapsBetweenComplexProjectiveSpaces'> <h6>Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><msub><mi>n</mi> <mn>1</mn></msub><mo>≤</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1 \leq n_1 \leq n_2</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d \in \mathbb{N}</annotation></semantics></math>, the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Maps</mi> <mi>rat</mi> <mrow><mi>deg</mi><mo>=</mo><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>Maps</mi> <mi>cts</mi> <mrow><mi>deg</mi><mo>=</mo><mi>d</mi></mrow></msubsup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> Maps^{deg = d}_{rat} \big( \mathbb{C}P^{n_1} ,\, \mathbb{C}P^{n_2} \big) \xhookrightarrow{\;\;\;\;} Maps^{deg = d}_{cts} \big( \mathbb{C}P^{n_1} ,\, \mathbb{C}P^{n_2} \big) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/topological+subspace">topological subspace</a> of <a class="existingWikiWord" href="/nlab/show/rational+maps">rational maps</a> of algebraic degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> of <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math> in the sense of <a class="maruku-eqref" href="#eq:DegreeOfAMapBetweenComplexProjectiveSpaces">(3)</a> induces an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> on integral <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> in degrees</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mspace width="thinmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">(</mo><mn>2</mn><msub><mi>n</mi> <mn>1</mn></msub><mo>−</mo><mn>2</mn><msub><mi>n</mi> <mn>2</mn></msub><mo>+</mo><mn>1</mn><mo maxsize="1.2em" minsize="1.2em">)</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">⌊</mo><mo stretchy="false">(</mo><mi>d</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">⌋</mo><mo>+</mo><mn>1</mn><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \leq\, \big( 2 n_1 - 2 n_2 + 1 \big) \big( \lfloor(d+2)/2\rfloor + 1 \big) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⌊</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⌋</mo></mrow><annotation encoding="application/x-tex">\lfloor - \rfloor</annotation></semantics></math> denotes the integer <a class="existingWikiWord" href="/nlab/show/floor">floor</a> of a rational number.</p> </div> </p> <p>(<a href="#Mostovoy06">Mostovoy 2006, Theorem 2</a>, with corrected proof in <a href="#Mostovoy12">Mostovoy 2012</a>)</p> <p>An analogous result for <a class="existingWikiWord" href="/nlab/show/real+projective+spaces">real projective spaces</a> is in <a href="#AdamaszekKozlowskiYamaguchi08">Adamaszek, Kozlowski & Yamaguchi 2008</a>.</p> <p>Prop. <a class="maruku-ref" href="#HomologyEquivalenceForMapsBetweenComplexProjectiveSpaces"></a> generalizes to the case that the <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msup><mi>P</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{C}P^{n_2}</annotation></semantics></math> is allowed to be any <a class="existingWikiWord" href="/nlab/show/smooth+variety">smooth</a> <a class="existingWikiWord" href="/nlab/show/toric+variety">toric variety</a> (<a href="#MostovoyMunguiaVillanueva12">Mostovoy & Munguia-Villanueva 2012</a>, <a href="#KozlowskiYamaguchi18">Kozlowski & Yamaguchi 2018</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+monopole">Yang-Mills monopole</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Oka+principle">Oka principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cohomotopy">Cohomotopy</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="maps_out_of_riemann_surfaces">Maps out of Riemann surfaces</h3> <p>The original theorem for <a class="existingWikiWord" href="/nlab/show/rational+maps">rational maps</a>/<a class="existingWikiWord" href="/nlab/show/continuous+maps">continuous maps</a> from compact <a class="existingWikiWord" href="/nlab/show/Riemann+surfaces">Riemann surfaces</a> to <a class="existingWikiWord" href="/nlab/show/complex+projective+spaces">complex projective spaces</a>:</p> <ul> <li id="Segal79"><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>The topology of spaces of rational functions</em>, Acta Math. Volume 143 (1979), 39-72 (<a href="https://projecteuclid.org/euclid.acta/1485890033">euclid:1485890033</a>)</li> </ul> <p>Further discussion:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Fred+Cohen">Fred Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Ralph+Cohen">Ralph Cohen</a>, <a class="existingWikiWord" href="/nlab/show/B.+M.+Mann">B. M. Mann</a>, <a class="existingWikiWord" href="/nlab/show/R.+J.+Milgram">R. J. Milgram</a>, <em>The topology of rational functions and divisors of surfaces</em>, Acta Math (1991) 166: 163 (<a href="https://doi.org/10.1007/BF02398886">doi:10.1007/BF02398886</a>)</p> </li> <li id="FriedlanderLawson97"> <p><a class="existingWikiWord" href="/nlab/show/Eric+M.+Friedlander">Eric M. Friedlander</a>, <a class="existingWikiWord" href="/nlab/show/H.+Blaine+Lawson">H. Blaine Lawson</a>, Section 5.C of: <em>Duality Relating Spaces of Algebraic Cocycles and Cycles</em>, Topology Volume 36, Issue 2, March 1997, Pages 533-565 (<a href="https://dornsife.usc.edu/assets/sites/1163/docs/Preprint_versionsPublications/10/projH.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ralph+L.+Cohen">Ralph L. Cohen</a>, <a class="existingWikiWord" href="/nlab/show/John+D.+S.+Jones">John D. S. Jones</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+B.+Segal">Graeme B. Segal</a>, <em>Stability for holomorphic spheres and Morse theory</em>, in: K. Grove, I. H. Madsen, E. K. Pedersen (eds.) <em>Geometry and Topology: Aarhus</em>, Contemporary Mathematics</p> <p>Volume: 258 (2000) (<a href="https://arxiv.org/abs/math/9904185">arXiv:math/9904185</a>, <a href="https://bookstore.ams.org/conm-258">ISBN:978-0-8218-2158-9</a>)</p> </li> <li id="Kamiyama07"> <p>Yasuhiko Kamiyama, <em>Remarks on spaces of real rational functions</em>, The Rocky Mountain Journal of Mathematics Vol. 37, No. 1 (2007), pp. 247-257 (<a href="https://www.jstor.org/stable/44239357">jstor:44239357</a>)</p> </li> </ul> <p>The analog for rational curves into <a class="existingWikiWord" href="/nlab/show/real+projective+spaces">real projective spaces</a>:</p> <ul> <li id="Mostovoy01"><a class="existingWikiWord" href="/nlab/show/Jacob+Mostovoy">Jacob Mostovoy</a>, <em>Spaces of Rational Loops on a Real Projective Space</em>, Transactions of the American Mathematical Society, Vol. 353, No. 5 (May, 2001), pp. 1959-1970 (<a href="https://www.jstor.org/stable/221802">jstor:221802</a>)</li> </ul> <p>Identification of the higher <a class="existingWikiWord" href="/nlab/show/homotopy+groups">homotopy groups</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Maps</mi> <mi>rat</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup><mo>,</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Maps_{rat}(\mathbb{C}P^1, \mathbb{C}P^1)</annotation></semantics></math>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Martin+A.+Guest">Martin A. Guest</a>, <a class="existingWikiWord" href="/nlab/show/Andrzej+Kozlowski">Andrzej Kozlowski</a>, M. Murayama, <a class="existingWikiWord" href="/nlab/show/Kohhei+Yamaguchi">Kohhei Yamaguchi</a>, <em>The homotopy type of the space of rational functions</em>, J. Math. Kyoto Univ. 35(4): 631-638 (1995) (<a href="https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-35/issue-4/The-homotopy-type-of-the-space-of-rational-functions/10.1215/kjm/1250518652.full">doi:10.1215/kjm/1250518652</a>)</li> </ul> <p>Identification of the full <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Maps</mi> <mi>rat</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup><mo>,</mo><mi>ℂ</mi><msup><mi>P</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Maps_{rat}(\mathbb{C}P^1, \mathbb{C}P^n)</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/configuration+space+of+points">configuration space of points</a>:</p> <ul> <li id="CohenShimamoto91"><a class="existingWikiWord" href="/nlab/show/Ralph+L.+Cohen">Ralph L. Cohen</a>, <a class="existingWikiWord" href="/nlab/show/Don+H.+Shimamoto">Don H. Shimamoto</a>, <em>Rational Functions, Labelled Configurations, and Hilbert Schemes</em>, Journal of the London Mathematical Socienty <strong>43</strong> 2 (1991) 509-528 (<a href="https://doi.org/10.1112/jlms/s2-43.3.509">doi:10.1112/jlms/s2-43.3.509</a>)</li> </ul> <p>Generalization to the case that the <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> is</p> <p>… a <a class="existingWikiWord" href="/nlab/show/Grassmannian">Grassmannian</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Frances+Kirwan">Frances Kirwan</a>, <em>On spaces of maps from Riemann surfaces to Grassmannians and applications to the cohomology of moduli of vector bundles</em>, Ark. Mat. 24(1-2): 221-275 (1985) (<a href="https://www.projecteuclid.org/journals/arkiv-for-matematik/volume-24/issue-1-2/On-spaces-of-maps-from-Riemann-surfaces-to-Grassmannians-and/10.1007/BF02384399.full">doi:10.1007/BF02384399</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Benjamin+M.+Mann">Benjamin M. Mann</a>, <a class="existingWikiWord" href="/nlab/show/R.+James+Milgram">R. James Milgram</a>, <em>Some spaces of holomorphic maps to complex Grassmann manifolds</em>, J. Differential Geom. 33(2): 301-324 (1991) (<a href="https://projecteuclid.org/journals/journal-of-differential-geometry/volume-33/issue-2/Some-spaces-of-holomorphic-maps-to-complex-Grassmann/10.4310/jdg/1214446318.full">doi:10.4310/jdg/1214446318</a>)</p> </li> <li id="Havlicek92"> <p>John W. Havlicek, <em>On spaces of holomorphic maps from two copies of the riemann sphere to complex grassmannians</em>, Stanford University 1992 (<a href="https://www.proquest.com/docview/304009877">proquest:304009877</a>)</p> </li> </ul> <p>… a <a class="existingWikiWord" href="/nlab/show/toric+variety">toric variety</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+A.+Guest">Martin A. Guest</a>, <em>Configuration spaces and the space of rational curves on a toric variety</em>, Bull. Amer. Math. Soc. 31 (1994), 191-196 (<a href="https://doi.org/10.1090/S0273-0979-1994-00515-4">doi:10.1090/S0273-0979-1994-00515-4</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Martin+A.+Guest">Martin A. Guest</a>, <em>The topology of the space of rational curves on a toric variety</em>, Acta Math. 174(1): 119-145 (1995) (<a href="https://projecteuclid.org/journals/acta-mathematica/volume-174/issue-1/The-topology-of-the-space-of-rational-curves-on-a/10.1007/BF02392803.full">doi:10.1007/BF02392803</a>, <a href="https://arxiv.org/abs/alg-geom/9301005">arXiv:alg-geom/9301005</a>)</p> </li> </ul> <p>… a <a class="existingWikiWord" href="/nlab/show/flag+manifold">flag manifold</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/C.+P.+Boyer">C. P. Boyer</a>, <a class="existingWikiWord" href="/nlab/show/B.+M.+Mann">B. M. Mann</a>, <a class="existingWikiWord" href="/nlab/show/J.+C.+Hurtubise">J. C. Hurtubise</a>, <a class="existingWikiWord" href="/nlab/show/R.+J.+Milgram">R. J. Milgram</a>, <em>The topology of the space of rational maps into generalized flag manifolds</em>, Acta Mathematica. 1994 Mar 1;173(1):61-101 (<a href="https://projecteuclid.org/journals/acta-mathematica/volume-173/issue-1/The-topology-of-the-space-of-rational-maps-into-generalized/10.1007/BF02392569.full">doi:10.1007/BF02392569</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/J.+C.+Hurtubise">J. C. Hurtubise</a>, <em>Holomorphic maps of a Riemann surface into a flag manifold</em>, J. Differential Geom. 43(1): 99-118 (1996) (<a href="https://projecteuclid.org/journals/journal-of-differential-geometry/volume-43/issue-1/Holomorphic-maps-of-a-Riemann-surface-into-a-flag-manifold/10.4310/jdg/1214457899.full">doi:10.4310/jdg/1214457899</a>)</p> </li> </ul> <p>Application to the <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> of <a class="existingWikiWord" href="/nlab/show/Skyrmions">Skyrmions</a> (via their rational map Ansatz, see the references <a href="skyrmion#SkyrmionsFromRationalMapsReferences">there</a>):</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em>Homotopy of rational maps and the quantization of Skyrmions</em>, Annals of Physics Volume 304, Issue 2, April 2003, Pages 103-127 (<a href="https://doi.org/10.1016/S0003-4916(03)00014-9">doi:10.1016/S0003-4916(03)00014-9</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em>Skyrmions and Rational Maps</em>, talk at KIAS 2004 (<a href="http://newton.kias.re.kr/KH04/talks/krusch.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Krusch_SkyrmionsAndRationalMaps.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steffen+Krusch">Steffen Krusch</a>, <em>Quantization of Skyrmions</em> (<a href="https://arxiv.org/abs/hep-th/0610176">arXiv:hep-th/0610176</a>)</p> </li> </ul> <h3 id="maps_between_projective_spaces_2">Maps between projective spaces</h3> <p>On maps between <a class="existingWikiWord" href="/nlab/show/complex+projective+spaces">complex projective spaces</a>:</p> <ul> <li id="Mostovoy06"><a class="existingWikiWord" href="/nlab/show/Jacob+Mostovoy">Jacob Mostovoy</a>, <em>Spaces of rational maps and the Stone–Weierstrass theorem</em>, Topology Volume 45, Issue 2, March 2006, Pages 281-293 (<a href="https://doi.org/10.1016/j.top.2005.08.003">doi:10.1016/j.top.2005.08.003</a>)</li> </ul> <p>corrected proof in:</p> <ul> <li id="Mostovoy12"><a class="existingWikiWord" href="/nlab/show/Jacob+Mostovoy">Jacob Mostovoy</a>, <em>Truncated Simplicial Resolutions and Spaces of Rational Maps</em>, The Quarterly Journal of Mathematics, Volume 63, Issue 1, March 2012, Pages 181–187 (<a href="https://doi.org/10.1093/qmath/haq031">doi:10.1093/qmath/haq031</a>)</li> </ul> <p>On maps between <a class="existingWikiWord" href="/nlab/show/real+projective+spaces">real projective spaces</a>:</p> <ul> <li id="AdamaszekKozlowskiYamaguchi08"><a class="existingWikiWord" href="/nlab/show/Michal+Adamaszek">Michal Adamaszek</a>, <a class="existingWikiWord" href="/nlab/show/Andrzej+Kozlowski">Andrzej Kozlowski</a>, <a class="existingWikiWord" href="/nlab/show/Kohhei+Yamaguchi">Kohhei Yamaguchi</a>, <em>Spaces of algebraic and continuous maps between real algebraic varieties</em>, Quart. J. Math. 62 (2011), 771–790 (<a href="https://arxiv.org/abs/0809.4893">arXiv:0809.4893</a>, <a href="https://doi.org/10.1093/qmath/haq029">doi:10.1093/qmath/haq029</a>)</li> </ul> <p>On maps from <a class="existingWikiWord" href="/nlab/show/real+projective+space">real projective space</a> to <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrzej+Kozlowski">Andrzej Kozlowski</a>, <a class="existingWikiWord" href="/nlab/show/Kohhei+Yamaguchi">Kohhei Yamaguchi</a>, <em>Spaces of algebraic maps from real projective spaces into complex projective spaces</em> (<a href="https://arxiv.org/abs/0812.3954">arXiv:0812.3954</a>), in: <a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a> et al. (eds.) <em>Homotopy Theory of Function Spaces and Related Topics</em>, Contemporary Mathematics Volume: 519; 2010 (<a href="https://bookstore.ams.org/conm-519">ISBN:978-0-8218-4929-3</a>)</li> </ul> <p>and <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariantly</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andrzej+Kozlowski">Andrzej Kozlowski</a>, <a class="existingWikiWord" href="/nlab/show/Kohhei+Yamaguchi">Kohhei Yamaguchi</a>, <em>Spaces of equivariant algebraic maps from real projective spaces into complex projective spaces</em>, RIMS Kôkyûroku Bessatsu B39 (2013), 051−061 (<a href="https://arxiv.org/abs/1109.0353">arXiv:1109.0353</a>, <a href="https://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B39/pdf/B39_006.pdf">published pdf</a>)</li> </ul> <h3 id="maps_from_projective_spaces_to_toric_varieties">Maps from projective spaces to toric varieties</h3> <p>On maps from <a class="existingWikiWord" href="/nlab/show/complex+projective+space">complex projective space</a> to smooth <a class="existingWikiWord" href="/nlab/show/toric+varieties">toric varieties</a>:</p> <ul> <li id="MostovoyMunguiaVillanueva12"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Mostovoy">Jacob Mostovoy</a>, Erendira Munguia-Villanueva, <em>Spaces of morphisms from a projective space to a toric variety</em>, Revista Colombiana de Matematicas <strong>48</strong> 1 (2014) (<a href="https://arxiv.org/abs/1210.2795">arXiv:1210.2795</a>, <a href="http://scm.org.co/archivos/revista/Articulos/1129.pdf">published pdf</a>)</p> </li> <li id="KozlowskiYamaguchi18"> <p><a class="existingWikiWord" href="/nlab/show/Andrzej+Kozlowski">Andrzej Kozlowski</a>, <a class="existingWikiWord" href="/nlab/show/Kohhei+Yamaguchi">Kohhei Yamaguchi</a>, <em>The homotopy type of spaces of rational curves on a toric variety</em>, Topology and its Applications Volume 249, 1 November 2018, Pages 19-42 (<a href="https://doi.org/10.1016/j.topol.2018.06.006">doi:10.1016/j.topol.2018.06.006</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 16, 2021 at 12:48:42. 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