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href="/nlab/show/stable+homotopy+theory">stable</a>) <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy</a></p> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> </li> </ul> </div></div> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#InOrdinaryCohomology'>In ordinary cohomology</a></li> <li><a href='#in_generalized_cohomology'>In generalized cohomology</a></li> <li><a href='#InRelativeCohomology'>In relative cohomology</a></li> <li><a href='#in_equivariant_cohomology'>In equivariant cohomology</a></li> </ul> <li><a href='#details'>Details</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#consequences'>Consequences</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#in_equivariant_cohomology_2'>In equivariant cohomology</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="InOrdinaryCohomology">In ordinary cohomology</h3> <p>The <em>Serre spectral sequence</em> or <em>Leray-Serre spectral sequence</em> is a <a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a> for computation of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> (<a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a>) of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> in a <a class="existingWikiWord" href="/nlab/show/Serre+fibration">Serre</a>-<a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>.</p> <p> <div class='num_prop' id='OrdinaryCohomologicalSerreSpectralSequence'> <h6>Proposition</h6> <p><strong>(ordinary cohomology Serre spectral sequence)</strong><br /></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F &\longrightarrow& E \\ && \downarrow^{\mathrlap{p}} \\ && X } </annotation></semantics></math></div> <p>over a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, such that the canonical <a class="existingWikiWord" href="/nlab/show/group+action">group action</a> of the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(X)</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/trivial+action">trivial</a> (for instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected topological space</a>), then there exists a <em>cohomology Serre spectral sequence</em> of the form:</p> <div class="maruku-equation" id="eq:SecondPageAndConvergenceOfOrdinaryCohomologicalSerreSpectralSequence"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mi>p</mi></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mrow><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_2^{p,q} \;=\; H^p \big( X, \, H^q(F) \big) \;\Rightarrow\; H^{p+q}(E) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>(e.g. <a href="#Hatcher">Hatcher, Thm. 1.14</a>)</p> <p id="ConvergenceOfTheOrdinaryCohomologySSS"> Hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> we have a <a class="existingWikiWord" href="/nlab/show/filtration">filtration</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a></p> <div class="maruku-equation" id="eq:FilteringForOrdinaryCohomologySerreSpectralSequence"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>n</mi><mo>,</mo><mn>0</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>F</mi> <mi>n</mi> <mi>n</mi></msubsup><mover><mo>↪</mo><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msubsup></mrow></mover><msubsup><mi>F</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>⋯</mi><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>F</mi> <mn>1</mn> <mi>n</mi></msubsup><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mn>0</mn><mo>,</mo><mi>n</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><msubsup><mi>F</mi> <mn>0</mn> <mi>n</mi></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> 0 \xhookrightarrow{ \;\; E^{n,0}_\infty \;\; } F^n_n \xhookrightarrow{ E^{n-1,1}_\infty } F^n_{n-1} \xhookrightarrow{\;\;\;\;\;} \cdots \xhookrightarrow{\;\;\;\;\;} F^n_{1} \xhookrightarrow{ \;\; E^{0,n}_\infty \;\; } F^n_0 \;=\; H^n(E) \,, </annotation></semantics></math></div> <p>where</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mover><mo>↪</mo><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi></mrow></msubsup></mrow></mover><msubsup><mi>F</mi> <mi>p</mi> <mi>n</mi></msubsup><mphantom><mi>AAAA</mi></mphantom><mtext>means that</mtext><mphantom><mi>AAAA</mi></mphantom><mn>0</mn><mo>→</mo><msubsup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mover><mo>↪</mo><mspace width="thickmathspace"></mspace></mover><msubsup><mi>F</mi> <mi>p</mi> <mi>n</mi></msubsup><mo>↠</mo><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi></mrow></msubsup><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> F^n_{p+1} \xhookrightarrow{ E^{p,n-p}_\infty } F^n_{p} {\phantom{AAAA}} \text{means that} {\phantom{AAAA}} 0 \to F^n_{p+1} \xhookrightarrow{\;} F^n_{p} \twoheadrightarrow E^{p,n-p}_\infty \to 0 \,, </annotation></semantics></math></div> <p>hence that – iteratively as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> <em>decreases</em> – <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>F</mi> <mi>p</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">F^n_p</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/algebra+extension">extension</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mi>p</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>p</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex">E^{p,n-p}_\infty</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>F</mi> <mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">F^n_{p+1}</annotation></semantics></math>.</p> <h3 id="in_generalized_cohomology">In generalized cohomology</h3> <p>The generalization of this from <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> to <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a> is the <em><a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a></em>, see there for details.</p> <h3 id="InRelativeCohomology">In relative cohomology</h3> <p>There are two kinds of <strong>relative Serre spectral sequences</strong>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>→</mo><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">F \to E \to X</annotation></semantics></math> as above and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \hookrightarrow X</annotation></semantics></math> a subspace, the induced restriction of the fibration</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>F</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F & \simeq & F \\ \downarrow && \downarrow \\ p^{-1}(A) &\longrightarrow& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ A &\hookrightarrow& X } </annotation></semantics></math></div> <p>induces a spectral sequence in <a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a> of the base space of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo>;</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E_2^{p,q} = H^p(X,A; H^q(F)) \;\Rightarrow\; H^\bullet(E, p^{-1}(A)) \,. </annotation></semantics></math></div> <p>(e.g. <a href="#Davis91">Davis 91, theorem 9.33</a>)</p> <p>Conversely, for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi><mo>′</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>F</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>E</mi><mo>′</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F' & \hookrightarrow & F \\ \downarrow && \downarrow \\ E' &\hookrightarrow& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ X &\hookrightarrow& X } </annotation></semantics></math></div> <p>a sub-fibration over the same base, then this induces a spectral sequence for <a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a> of the the total space in terms of ordinary cohomology with coefficients in the relative cohomology of the fibers:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>;</mo><msup><mi>H</mi> <mi>q</mi></msup><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>F</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>E</mi><mo>′</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{p,q}_2 = H^p(X; H^q(F,F')) \;\Rightarrow\; H^\bullet(E,E') \,. </annotation></semantics></math></div> <p>(e.g. <a href="#Kochman96">Kochman 96, theorem 2.6.3</a>, <a href="#Davis91">Davis 91, theorem 9.34</a>)</p> <h3 id="in_equivariant_cohomology">In equivariant cohomology</h3> <p>There is also a generalization to <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a>: for <a href="Mackey+functor#Cohomology">cohomology with coefficients in a Mackey functor</a> with<a class="existingWikiWord" href="/nlab/show/RO%28G%29-grading">RO(G)-grading</a> for <a class="existingWikiWord" href="/nlab/show/representation+spheres">representation spheres</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>V</mi></msup></mrow><annotation encoding="application/x-tex">S^V</annotation></semantics></math>, then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E \to X</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-fibration of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a> and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Mackey+functor">Mackey functor</a>, the equivariant Serre spectral sequence looks like (<a href="#Kronholm10">Kronholm 10, theorem 3.1</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msubsup><mo>=</mo><msup><mi>H</mi> <mi>p</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>H</mi> <mrow><mi>V</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>F</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>⇒</mo><mspace width="thinmathspace"></mspace><msup><mi>H</mi> <mrow><mi>V</mi><mo>+</mo><mi>p</mi><mo>+</mo><mi>q</mi></mrow></msup><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> E_2^{p,q} = H^p(X, H^{V+q}(F,A)) \,\Rightarrow\, H^{V+p+q}(E,A) \,, </annotation></semantics></math></div> <p>where on the left in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math>-page we have <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the genuine equivariant cohomology groups of the fiber.</p> <h2 id="details">Details</h2> <p>For details on the plain Serre spectral sequence see at <em><a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a></em> and take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>H</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">E = H R</annotation></semantics></math> to be ordinary cohomology.</p> <h2 id="examples">Examples</h2> <p> <div class='num_remark' id='IntegralCohomologyOfHomotopyQuotientOf4SphereByADEAction'> <h6>Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> of <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> by <a class="existingWikiWord" href="/nlab/show/finite+subgroup+of+SU%282%29">finite subgroup of SU(2)</a>)</strong> <br /></p> <p>Let</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mover><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace></mrow></mover><mi>Sp</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>SU</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>Spin</mi><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \xhookrightarrow{\;i\;} Sp(1) \simeq SU(2) \simeq Spin(3)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/finite+subgroup+of+SU%282%29">finite subgroup of SU(2)</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>G</mi><mi>Actions</mi><mo stretchy="false">(</mo><msub><mi>VectorSpaces</mi> <mi>ℝ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{H} \,\in\, G Actions(VectorSpaces_{\mathbb{R}})</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a>, regarded as a 4d <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a>, which is the <a class="existingWikiWord" href="/nlab/show/restricted+representation">restricted representation</a> of the defining representation of <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>ℍ</mi></msup><mo>∈</mo><mi>G</mi><mi>Actions</mi><mo stretchy="false">(</mo><mi>TopologicalSpaces</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S^{\mathbb{H}} \in G Actions(TopologicalSpaces)</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a>.</p> </li> </ul> <p>Then the <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> in degree 4 of the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> is the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> <div class="maruku-equation" id="eq:Integral4CohomologyOfHomotopyQuotientOf4SphereByFiniteSubgroupOfSU2"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mi>ℍ</mi></msup><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo>⊕</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^4 \big( S^{\mathbb{H}} \!\sslash\! G, \, \mathbb{Z} \big) \;\; \simeq \;\; \mathbb{Z} \oplus (\mathbb{Z}/\left\vert G \right\vert) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> with the <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a> of <a class="existingWikiWord" href="/nlab/show/order+of+a+group">order</a> that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>By the <a class="existingWikiWord" href="/nlab/show/Borel+construction">Borel construction</a> we have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of the form</p> <div class="maruku-equation" id="eq:BorelConstructionFor4SphereActedOnByADESubgroup"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>S</mi> <mn>4</mn></msup></mtd> <mtd><mo>⟶</mo></mtd> <mtd><msup><mi>S</mi> <mi>ℍ</mi></msup><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ S^4 &\longrightarrow& S^{\mathbb{H}} \!\sslash\! G \\ && \big\downarrow \\ && B G } </annotation></semantics></math></div> <p>over the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>Here the <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/4-sphere">4-sphere</a> <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> is (e.g. by the nature of the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbb{Z},4)</annotation></semantics></math>)</p> <div class="maruku-equation" id="eq:IntegralCohomologyOfThe4Sphere"><span class="maruku-eq-number">(5)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mi>for</mi></mtd> <mtd><mi>n</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>4</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mtext>otherwise</mtext><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> H^n \big( S^4, \, \mathbb{Z} \big) \;\simeq\; \left\{ \array{ \mathbb{Z} & for & n \in \{0,4\} \\ 0 & \text{otherwise} \,. } \right. </annotation></semantics></math></div> <p>We claim that the <a class="existingWikiWord" href="/nlab/show/group+action">group action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\pi_1(B G) \simeq G</annotation></semantics></math> (by <a href="simplicial+classifying+space#HomotopyGroupsOfBarWG">this Prop.</a>) on the <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> is <a class="existingWikiWord" href="/nlab/show/trivial+action">trivial</a>. This follows by observing that:</p> <ol> <li> <p>we have an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a> between the <a class="existingWikiWord" href="/nlab/show/representation+sphere">representation sphere</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{H}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/unit+sphere">unit sphere</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \oplus \mathbb{H}</annotation></semantics></math> (by <a href="representation+sphere#RepresentationSpheresAsUnitSpheres">this Prop.</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>ℍ</mi></msup><mspace width="thickmathspace"></mspace><msub><mo>≃</mo> <mi>G</mi></msub><mspace width="thickmathspace"></mspace><mi>S</mi><mo stretchy="false">(</mo><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> S^{\mathbb{H}} \;\simeq_{G}\; S(\mathbb{R} \oplus \mathbb{H}) \,; </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/group+action">group action</a> of <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℍ</mi><msub><mo>≃</mo> <mi>ℝ</mi></msub><msup><mi>ℝ</mi> <mn>4</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{H} \simeq_{\mathbb{R}} \mathbb{R}^4</annotation></semantics></math> is through the defining action of <a class="existingWikiWord" href="/nlab/show/SO%284%29">SO(4)</a>, hence the action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>⊕</mo><mi>ℍ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \oplus \mathbb{H}</annotation></semantics></math> is through <a class="existingWikiWord" href="/nlab/show/SO%285%29">SO(5)</a>,</p> <p>because <a class="existingWikiWord" href="/nlab/show/quaternions">quaternions</a> are a <a class="existingWikiWord" href="/nlab/show/normed+division+algebra">normed division algebra</a>, so that left-multiplication by unit-<a class="existingWikiWord" href="/nlab/show/norm">norm</a> quaternions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">q \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Sp%281%29">Sp(1)</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><mi>ℍ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">= S(\mathbb{H})</annotation></semantics></math> preserves the <a class="existingWikiWord" href="/nlab/show/norm">norm</a> (e.g <a class="existingWikiWord" href="/schreiber/show/Equivariant+homotopy+and+super+M-branes">HSS 18, Rem. A.8</a>);</p> </li> <li> <p>the generator of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^4(S^4,\mathbb{Z})</annotation></semantics></math> may be identified with the <a class="existingWikiWord" href="/nlab/show/volume+form">volume form</a> (under the <a class="existingWikiWord" href="/nlab/show/Hopf+degree+theorem">Hopf degree theorem</a> and the <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>) which is manifestly preserved by the action of the <a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SO</mi><mo stretchy="false">(</mo><mn>5</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SO(5)</annotation></semantics></math>.</p> </li> </ol> <p>Therefore, the <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral</a>-cohomological Serre spectral sequence (Prop. <a class="maruku-ref" href="#OrdinaryCohomologicalSerreSpectralSequence"></a>) applies to the Borel fiber sequence <a class="maruku-eqref" href="#eq:BorelConstructionFor4SphereActedOnByADESubgroup">(4)</a>.</p> <p>Now, noticing that the <a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a> of a <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> is its <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msubsup><mi>H</mi> <mi>grp</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^\bullet(B G, \mathbb{Z}) \;\simeq\; H^\bullet_{grp}(G, \mathbb{Z}) </annotation></semantics></math></div> <p>we have for the given <a class="existingWikiWord" href="/nlab/show/finite+subgroup+of+SU%282%29">finite subgroup of SU(2)</a> (by <a href="finite+rotation+group#GroupCohomologyOfFiniteSubgroupsOfSU2">this Prop</a>) that:</p> <div class="maruku-equation" id="eq:IntegralCohomologyOfClassifyingSpaceOfFiniteSubgroupOfSU2"><span class="maruku-eq-number">(6)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>ℤ</mi></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mi>G</mi> <mi>ab</mi></msup></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mo>=</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>4</mn></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">/</mo><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>n</mi><mspace width="thinmathspace"></mspace><mtext>positive multiple of</mtext><mspace width="thinmathspace"></mspace><mn>4</mn></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mtext>otherwise</mtext><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> H^n(B G, \mathbb{Z}) \;\simeq\; \left\{ \array{ \mathbb{Z} &\vert& n = 0 \\ G^{ab} &\vert& n = 2 \, mod \, 4 \\ \mathbb{Z}/{\vert G \vert} &\vert& n \, \text{positive multiple of} \, 4 \\ 0 &\vert& \text{otherwise} \,, } \right. </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>G</mi> <mi>ab</mi></msup><mo>≔</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">[</mo><mi>G</mi><mo>,</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">G^{ab} \coloneqq G / [G,G]</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/abelianization">abelianization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> <p>Using the cohomology groups <a class="maruku-eqref" href="#eq:IntegralCohomologyOfThe4Sphere">(5)</a> and <a class="maruku-eqref" href="#eq:IntegralCohomologyOfClassifyingSpaceOfFiniteSubgroupOfSU2">(6)</a> in the fomula <a class="maruku-eqref" href="#eq:SecondPageAndConvergenceOfOrdinaryCohomologicalSerreSpectralSequence">(1)</a> for the second page <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>2</mn> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">E_2^{\bullet, \bullet}</annotation></semantics></math> of the cohomology Serre spectral sequence (Prop. <a class="maruku-ref" href="#OrdinaryCohomologicalSerreSpectralSequence"></a>) shows that this is of the following form:</p> <svg 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92.544)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(75%, 75%, 75%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.488279 2.867271 C -2.031248 1.148521 -1.019529 0.336021 0.0000025 0.00008375 C -1.019529 -0.335854 -2.031248 -1.148354 -2.488279 -2.87101 " transform="matrix(1, 0, 0, -1, 232.82031, 161.78524)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wtZlSQ5dYHFoeUgXFu_rb0A95Eo=-glyph-6-2" x="215.24" y="170.044"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wtZlSQ5dYHFoeUgXFu_rb0A95Eo=-glyph-6-1" x="222.017773" y="170.044"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(75%, 75%, 75%)" stroke-opacity="1" stroke-miterlimit="10" d="M -100.690062 -83.596625 L -100.690062 83.118219 " transform="matrix(1, 0, 0, -1, 121.776, 92.544)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(75%, 75%, 75%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.484675 2.86758 C -2.03155 1.14883 -1.019831 0.33633 -0.0003 0.0003925 C -1.019831 -0.335545 -2.03155 -1.148045 -2.484675 -2.870701 " transform="matrix(0, -1, -1, 0, 21.08633, 9.1872)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wtZlSQ5dYHFoeUgXFu_rb0A95Eo=-glyph-6-3" x="8.528" y="23.986"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#wtZlSQ5dYHFoeUgXFu_rb0A95Eo=-glyph-7-1" x="14.124" y="21.174"></use> </g> </svg> <p>Since the <a class="existingWikiWord" href="/nlab/show/codomains">codomains</a> of the <a class="existingWikiWord" href="/nlab/show/differentials">differentials</a> on all the following pages are translated diagonally (downwards and rightwards, by the <a href="spectral+sequence#eq:FormOfDifferentialInCohomologySpectralSequence">general formula</a>) from the codomains seen above, one sees that for every differential on every page, the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> or the <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> is the <a class="existingWikiWord" href="/nlab/show/zero+group">zero group</a>.</p> <p>This means that all differentials are the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a>, hence that the spectral sequence collapses already on this second page:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>E</mi> <mn>2</mn> <mrow><mo>•</mo><mo>,</mo><mo>•</mo></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^{\bullet, \bullet}_\infty \;\; \simeq \;\; E^{\bullet, \bullet}_2 \,. </annotation></semantics></math></div> <p>Therefore the convergence statement <a class="maruku-eqref" href="#eq:FilteringForOrdinaryCohomologySerreSpectralSequence">(2)</a> says that the degree-4 cohomology group in question is a <a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mn>0</mn><mo>,</mo><mn>4</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mn>0</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mn>0</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi></mrow><annotation encoding="application/x-tex"> E^{0,4}_\infty \;\simeq\; H^0\big(B G, \, H^4(S^4; \mathbb{Z}) \big) \;\simeq\; H^0\big( B G, \, \mathbb{Z}\big) \;\simeq\; \mathbb{Z} </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>E</mi> <mn>∞</mn> <mrow><mn>4</mn><mo>,</mo><mn>0</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mn>4</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mo>;</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mn>4</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>B</mi><mi>G</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></mrow><annotation encoding="application/x-tex"> E^{4,0}_\infty \;\simeq\; H^4\big(B G, \, H^0(S^4; \mathbb{Z}) \big) \;\simeq\; H^4\big( B G, \, \mathbb{Z}\big) \;\simeq\; \mathbb{Z}/\left\vert G \right\vert </annotation></semantics></math></div> <p>in that we have a <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>↪</mo><msup><mi>H</mi> <mn>4</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>↠</mo><mi>ℤ</mi><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to \mathbb{Z}/\left\vert G\right\vert \hookrightarrow H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big) \twoheadrightarrow \mathbb{Z} \to 0 \,. </annotation></semantics></math></div> <p>But since the <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a> of the <a class="existingWikiWord" href="/nlab/show/integers">integers</a> is <a class="existingWikiWord" href="/nlab/show/trivial+group">trivial</a> (<a href="Ext#ExtensionsOfTheIntegersAreTrivial">this Expl.</a>) this extension must be the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>S</mi> <mn>4</mn></msup><mspace width="negativethinmathspace"></mspace><mo>⫽</mo><mspace width="negativethinmathspace"></mspace><mi>G</mi><mo>;</mo><mspace width="thinmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>ℤ</mi> <mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow></msub><mo>⊕</mo><mi>ℤ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^4\big( S^4 \!\sslash\! G;\, \mathbb{Z} \big) \;\simeq\; \mathbb{Z}_{\left\vert G\right\vert} \oplus \mathbb{Z} \,. </annotation></semantics></math></div> <p>This is the claim <a class="maruku-eqref" href="#eq:Integral4CohomologyOfHomotopyQuotientOf4SphereByFiniteSubgroupOfSU2">(3)</a> to be proven.</p> </div> </p> <h2 id="consequences">Consequences</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+long+exact+sequence">Serre long exact sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom-Gysin+sequence">Thom-Gysin sequence</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+generalized+cohomology">long exact sequence in generalized cohomology</a></li> </ul> </li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <p>The original article is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean-Pierre+Serre">Jean-Pierre Serre</a>, <em>Homologie singuliére des espaces fibrés</em> Applications, Ann. of Math. 54 (1951),</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Hatcher"> <p><a class="existingWikiWord" href="/nlab/show/Alan+Hatcher">Alan Hatcher</a>, <em><a href="https://pi.math.cornell.edu/~hatcher/SSAT/SSATpage.html">Spectral sequences in algebraic topology</a></em> <em>I: The Serre spectral sequence</em> (<a href="https://pi.math.cornell.edu/~hatcher/SSAT/SSch1.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Hatcher_SerreSpectralSequence.pdf" title="pdf">pdf</a>)</p> </li> <li id="Kochman96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochman">Stanley Kochman</a>, section 2.2. of <em><a class="existingWikiWord" href="/nlab/show/Bordism%2C+Stable+Homotopy+and+Adams+Spectral+Sequences">Bordism, Stable Homotopy and Adams Spectral Sequences</a></em>, AMS 1996</p> </li> <li id="Davis91"> <p>Davis, <em>Lecture notes in algebraic topology</em>, 1991</p> </li> </ul> <p>Lecture notes etc. includes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Greg+Friedman">Greg Friedman</a>, <em>Some extremely brief notes on the Leray spectral sequence</em> (<a href="http://faculty.tcu.edu/richardson/Seminars/Gregspecseq.pdf">pdf</a>)</li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> includes</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em><a href="http://homotopytypetheory.org/2013/08/08/spectral-sequences/">Spectral Sequences</a></em>, 2013</li> </ul> <p>and implementation in <a class="existingWikiWord" href="/nlab/show/Lean">Lean</a> is in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Floris+van+Doorn">Floris van Doorn</a>, <a class="existingWikiWord" href="/nlab/show/Egbert+Rijke">Egbert Rijke</a>, <a class="existingWikiWord" href="/nlab/show/Ulrik+Buchholtz">Ulrik Buchholtz</a>, <a class="existingWikiWord" href="/nlab/show/Favonia">Favonia</a>, <a class="existingWikiWord" href="/nlab/show/Steve+Awodey">Steve Awodey</a>, <a class="existingWikiWord" href="/nlab/show/Jeremy+Avigad">Jeremy Avigad</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <a class="existingWikiWord" href="/nlab/show/Jonas+Frey">Jonas Frey</a>, <em>Spectral</em> (<a href="https://github.com/cmu-phil/Spectral">github.com/cmu-phil/Spectral</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Floris+van+Doorn">Floris van Doorn</a>, §5 in: <em>On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory</em> (2018) [<a href="https://arxiv.org/abs/1808.10690">arXiv:1808.10690</a>]</p> </li> </ul> <h3 id="in_equivariant_cohomology_2">In equivariant cohomology</h3> <p>In <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a>, for <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, J.-A. Svensson, <em>The Equivariant Serre Spectral Sequence</em>, Proceedings of the American Mathematical Society Vol. 118, No. 1 (May, 1993), pp. 263-278 (<a href="http://www.jstor.org/stable/2160037">JSTOR</a>)</li> </ul> <p>and for genuine equivariant cohomology, i.e. for <a class="existingWikiWord" href="/nlab/show/RO%28G%29-grading">RO(G)</a>-graded <a href="Mackey+functor#Cohomology">cohomology with coefficients in a Mackey functor</a>:</p> <ul> <li id="Kronholm10"><a class="existingWikiWord" href="/nlab/show/William+C.+Kronholm">William C. Kronholm</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RO</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">RO(G)</annotation></semantics></math>-graded Serre spectral sequence</em>, Homology Homotopy Appl. <strong>12</strong> 1 (2010) 75-92. [<a href="https://arxiv.org/abs/0908.3827">arXiv:0908.3827</a>, <a href="https://doi.org/10.4310/HHA.2010.v12.n1.a7">doi:10.4310/HHA.2010.v12.n1.a7</a>, <a href="https://projecteuclid.org/euclid.hha/1296223823">euclid:hha/1296223823</a>]</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Megan+Shulman">Megan Shulman</a>, <em>Equivariant Spectral Sequences for Local Coefficients</em> (<a href="http://arxiv.org/abs/1005.0379">arXiv:1005.0379</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 13, 2023 at 13:29:51. 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