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href="/nlab/show/homology">homology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/chain">chain</a>, <a class="existingWikiWord" href="/nlab/show/cycle">cycle</a>, <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/characteristic+class">characteristic class</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+characteristic+class">universal characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>/<a class="existingWikiWord" href="/nlab/show/long+exact+sequence+in+cohomology">long exact sequence in cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/twisted+%E2%88%9E-bundle">twisted ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/obstruction">obstruction</a></p> </li> </ul> <h3 id="special_and_general_types">Special and general types</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+group+cohomology">nonabelian group cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group+cohomology">Lie group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+cohomology">Galois cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid+cohomology">groupoid cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+groupoid+cohomology">nonabelian groupoid cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+cohomology+theory">cobordism cohomology theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+cohomology">integral cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/taf">taf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+cohomology">Dolbeault cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a>, <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a>, <a class="existingWikiWord" href="/nlab/show/Brauer+group">Brauer group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/syntomic+cohomology">syntomic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivic+cohomology">motivic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+of+operads">cohomology of operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+cohomology">cyclic cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+topology">string topology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>, <a class="existingWikiWord" href="/nlab/show/groupal+model+for+universal+principal+%E2%88%9E-bundles">groupal model for universal principal ∞-bundles</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>/<a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+constant+%E2%88%9E-stack">covering ∞-bundle</a>/<a class="existingWikiWord" href="/nlab/show/local+system">local system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-vector+bundle">(∞,1)-vector bundle</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-vector+bundle">(∞,n)-vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/Spin+structure">Spin structure</a>, <a class="existingWikiWord" href="/nlab/show/Spin%5Ec+structure">Spin^c structure</a>, <a class="existingWikiWord" href="/nlab/show/String+structure">String structure</a>, <a class="existingWikiWord" href="/nlab/show/Fivebrane+structure">Fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+with+constant+coefficients">cohomology with constant coefficients</a> / <a class="existingWikiWord" href="/nlab/show/cohomology+with+a+local+system+of+coefficients">with a local system of coefficients</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+extensions">Lie algebra extensions</a>, <a class="existingWikiWord" href="/nlab/show/Gelfand-Fuks+cohomology">Gelfand-Fuks cohomology</a>,</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gerstenhaber-Schack+cohomology">bialgebra cohomology</a></p> </li> </ul> <h3 id="special_notions">Special notions</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%3Fech+cohomology">?ech cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hypercohomology">hypercohomology</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+cohomology">twisted cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+bundle">twisted bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin+structure">twisted spin structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+spin%5Ec+structure">twisted spin^c structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/twisted+differential+c-structures">twisted differential c-structures</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twisted+differential+string+structure">twisted differential string structure</a>, <a class="existingWikiWord" href="/nlab/show/twisted+differential+fivebrane+structure">twisted differential fivebrane structure</a></li> </ul> </li> </ul> </li> <li> <p>differential cohomology</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential generalized (Eilenberg-Steenrod) cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative cohomology</a></p> </li> </ul> <h3 id="extra_structure">Extra structure</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/orientation+in+generalized+cohomology">in generalized cohomology</a></p> </li> </ul> <h3 id="operations">Operations</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration">fiber integration</a>, <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+localization">cohomology localization</a></p> </li> </ul> <h3 id="theorems">Theorems</h3> <ul> <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> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a>, <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a>, <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+theory">Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+theorem">Hodge theorem</a></p> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Hodge+theory">nonabelian Hodge theory</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+Hodge+theory">noncommutative Hodge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">hypercovering theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+duality">Eckmann-Hilton-Fuks duality</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#reduced_homology'>Reduced homology</a></li> <li><a href='#unreduced_homology'>Unreduced homology</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#expression_by_ordinary_homology_via_atiyahhirzebruch_spectral_sequence'>Expression by ordinary homology via Atiyah-Hirzebruch spectral sequence</a></li> <li><a href='#WhiteheadTheorem'>Whitehead theorem</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>generalized homology theory</em> is a certain <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from suitable <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> to <a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a> which satisfies most, but not all, of the abstract properties of <a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> functors (e.g. <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a>).</p> <p>By the <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a>, under certain conditions every <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is the coefficient object of a <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized</a> <a class="existingWikiWord" href="/nlab/show/cohomology+theory">cohomology theory</a> and <a class="existingWikiWord" href="/nlab/show/Spanier-Whitehead+duality">S-dually</a> of a generalized homology theory. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mi>H</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">K = H R</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> this reduces to <a class="existingWikiWord" href="/nlab/show/homology">ordinary homology</a>.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></em> for more.</p> <h2 id="definition">Definition</h2> <h3 id="reduced_homology">Reduced homology</h3> <p>Throughout, write <a class="existingWikiWord" href="/nlab/show/Top">Top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>CW</mi></msub></mrow><annotation encoding="application/x-tex">{}_{CW}</annotation></semantics></math> for the category of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to <a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex">Top^{\ast/}_{CW}</annotation></semantics></math> for the corresponding category of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a>.</p> <p>Recall that <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Top</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex">Top^{\ast/}</annotation></semantics></math> are computed as colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math> after adjoining the base point and its inclusion maps to the given diagram</p> <div class="num_example" id="WedgeSumAsCoproduct"> <h6 id="example">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> is the <em><a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a></em>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\vee_{i \in I} X_i</annotation></semantics></math>.</p> </div> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>≔</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>∧</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo>⟶</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> \Sigma \coloneqq S^1 \wedge (-) \;\colon\; Top^{\ast/}_{CW} \longrightarrow Top^{\ast/}_{CW} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> functor.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^{\mathbb{Z}}</annotation></semantics></math> for the category of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a>.</p> <div class="num_defn" id="ReducedGeneralizedHomology"> <h6 id="definition_2">Definition</h6> <p>A <strong>reduced homology theory</strong> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex"> \tilde E_\bullet \;\colon\; (Top^{\ast/}_{CW}) \longrightarrow Ab^{\mathbb{Z}} </annotation></semantics></math></div> <p>from the category of <a class="existingWikiWord" href="/nlab/show/pointed+topological+spaces">pointed topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a> (“<a class="existingWikiWord" href="/nlab/show/homology+groups">homology groups</a>”), in components</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mover><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \tilde E _\bullet \;\colon\; (X \stackrel{f}{\longrightarrow} Y) \mapsto (\tilde E_\bullet(X) \stackrel{f_\ast}{\longrightarrow} \tilde E_\bullet(Y)) \,, </annotation></semantics></math></div> <p>and equipped with a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> of degree +1, to be called the <strong><a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a></strong>, of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \sigma \;\colon\; \tilde E_\bullet(-) \overset{\simeq}{\longrightarrow} \tilde E_{\bullet +1}(\Sigma -) </annotation></semantics></math></div> <p>such that:</p> <ol> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homotopy+invariance">homotopy invariance</a>)</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f_1,f_2 \colon X \longrightarrow Y</annotation></semantics></math> are two morphisms of pointed topological spaces such that there is a (base point preserving) <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>≃</mo><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_1 \simeq f_2</annotation></semantics></math> between them, then the induced <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of abelian groups are <a class="existingWikiWord" href="/nlab/show/equality">equal</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><msub><mo></mo><mo>*</mo></msub><mo>=</mo><msub><mi>f</mi> <mn>2</mn></msub><msub><mo></mo><mo>*</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f_1_\ast = f_2_\ast \,. </annotation></semantics></math></div></li> <li id="ReducedExactnessAxiom"> <p><strong>(exactness)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i \colon A \hookrightarrow X</annotation></semantics></math> an inclusion of pointed topological spaces, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>⟶</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j \colon X \longrightarrow Cone(i)</annotation></semantics></math> the induced <a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a>, then this gives an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a> of graded abelian groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>i</mi> <mo>*</mo></msub></mrow></mover><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>j</mi> <mo>*</mo></msub></mrow></mover><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>Cone</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tilde E_\bullet(A) \overset{i_\ast}{\longrightarrow} \tilde E_\bullet(X) \overset{j_\ast}{\longrightarrow} \tilde E_\bullet(Cone(i)) \,. </annotation></semantics></math></div></li> </ol> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\tilde E_\bullet</annotation></semantics></math> is <strong>additive</strong> if in addition</p> <ul> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/wedge+axiom">wedge axiom</a>)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>X</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{X_i\}_{i \in I} </annotation></semantics></math> any set of pointed CW-complexes, then the canonical morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>⊕</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><msup><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msup><mo stretchy="false">(</mo><msub><mo>∨</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \oplus_{i \in I} \tilde E_\bullet(X_i) \longrightarrow \tilde E^\bullet(\vee_{i \in I} X_i) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the value on the summands to the value on the <a class="existingWikiWord" href="/nlab/show/wedge+sum">wedge sum</a>, example <a class="maruku-ref" href="#WedgeSumAsCoproduct"></a>, is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </li> </ul> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\tilde E_\bullet</annotation></semantics></math> is <strong>ordinary</strong> if its value on the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">S^0</annotation></semantics></math> is concentrated in degree 0:</p> <ul> <li><strong>(Dimension)</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo>≠</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><msup><mi>𝕊</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\tilde E_{\bullet\neq 0}(\mathbb{S}^0) \simeq 0</annotation></semantics></math>.</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of reduced cohomology theories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo>⟶</mo><msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; \tilde E_\bullet \longrightarrow \tilde F_\bullet </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> between the underlying functors which is compatible with the suspension isomorphisms in that all the following <a class="existingWikiWord" href="/nlab/show/commuting+square">squares commute</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><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>σ</mi> <mi>E</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>σ</mi> <mi>F</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mover><mi>E</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mi>Σ</mi><mi>X</mi></mrow></msub></mrow></mover></mtd> <mtd><msub><mover><mi>F</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde E_\bullet(X) &\overset{\eta_X}{\longrightarrow}& \tilde F_\bullet(X) \\ {}^{\mathllap{\sigma_E}}\downarrow && \downarrow^{\mathrlap{\sigma_F}} \\ \tilde E_{\bullet + 1}(\Sigma X) &\overset{\eta_{\Sigma X}}{\longrightarrow}& \tilde F_{\bullet + 1}(\Sigma X) } \,. </annotation></semantics></math></div></div> <h3 id="unreduced_homology">Unreduced homology</h3> <p>In the following a <em>pair</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,A)</annotation></semantics></math> refers to a <a class="existingWikiWord" href="/nlab/show/subspace">subspace</a> inclusion of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> (<a class="existingWikiWord" href="/nlab/show/CW-complexes">CW-complexes</a>) <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>. Whenever only one space is mentioned, the subspace is assumed to be the <a class="existingWikiWord" href="/nlab/show/empty+set">empty set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, \emptyset)</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Top</mi> <mi>CW</mi> <mo>↪</mo></msubsup></mrow><annotation encoding="application/x-tex">Top_{CW}^{\hookrightarrow}</annotation></semantics></math> for the category of such pairs (the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>CW</mi></msub></mrow><annotation encoding="application/x-tex">Top_{CW}</annotation></semantics></math> on the inclusions). We identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Top</mi> <mi>CW</mi></msub><mo>↪</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mo>↪</mo></msubsup></mrow><annotation encoding="application/x-tex">Top_{CW} \hookrightarrow Top_{CW}^{\hookrightarrow}</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>↦</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \mapsto (X,\emptyset)</annotation></semantics></math>.</p> <div class="num_defn" id="GeneralizedHomologyTheory"> <h6 id="definition_3">Definition</h6> <p>A <strong>homology theory</strong> (unreduced, <a class="existingWikiWord" href="/nlab/show/relative+cohomology">relative</a>) is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub><mo>:</mo><mo stretchy="false">(</mo><msubsup><mi>Top</mi> <mi>CW</mi> <mo>↪</mo></msubsup><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>Ab</mi> <mi>ℤ</mi></msup></mrow><annotation encoding="application/x-tex"> E_\bullet : (Top_{CW}^{\hookrightarrow}) \longrightarrow Ab^{\mathbb{Z}} </annotation></semantics></math></div> <p>to the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+abelian+groups">graded abelian groups</a>, as well as a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> of degree +1, to be called the <strong><a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></strong>, of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \delta_{(X,A)} \;\colon\; E_{\bullet + 1}(X, A) \longrightarrow E_\bullet(A, \emptyset) \,. </annotation></semantics></math></div> <p>such that:</p> <ol> <li> <p><strong>(homotopy invariance)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \colon (X_1,A_1) \to (X_2,A_2)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> of pairs, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E_\bullet(f) \;\colon\; E_\bullet(X_1,A_1) \stackrel{\simeq}{\longrightarrow} E_\bullet(X_2,A_2) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>;</p> </li> <li id="ExactnessUnreduced"> <p><strong>(exactness)</strong> For <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> the induced sequence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>E</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>δ</mi></mover><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \to E_{n+1}(X, A) \stackrel{\delta}{\longrightarrow} E_n(A) \longrightarrow E_n(X) \longrightarrow E_n(X, A) \to \cdots </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>.</p> </li> <li id="excision"> <p><strong>(<a class="existingWikiWord" href="/nlab/show/excision">excision</a>)</strong> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↪</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \hookrightarrow A \hookrightarrow X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo>¯</mo></mover><mo>⊂</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\overline{U} \subset Int(A)</annotation></semantics></math>, then the natural inclusion of the pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i \colon (X-U, A-U) \hookrightarrow (X, A)</annotation></semantics></math> induces an isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E_\bullet(i) \;\colon\; E_n(X-U, A-U) \overset{\simeq}{\longrightarrow} E_n(X, A) </annotation></semantics></math></div></li> </ol> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">E_\bullet</annotation></semantics></math> is <strong>additive</strong> if it takes <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> to <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a>:</p> <ul> <li> <p><strong>(additivity)</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>i</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X, A) = \coprod_i (X_i, A_i)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>, then the canonical comparison morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>⊕</mo> <mi>i</mi></msub><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>E</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \oplus_i E_n(X_i, A_i) \overset{\simeq}{\longrightarrow} E_n(X, A) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>from the <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of the value on the summands, to the value on the total pair.</p> </li> </ul> <p>We say <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">E_\bullet</annotation></semantics></math> is <strong>ordinary</strong> if its value on the point is concentrated in degree 0</p> <ul> <li><strong>(Dimension)</strong>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mrow><mo>•</mo><mo>≠</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E_{\bullet \neq 0}(\ast,\emptyset) = 0</annotation></semantics></math>.</li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of unreduced homology theories</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mo>•</mo></msub><mo>⟶</mo><msub><mi>F</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; E_\bullet \longrightarrow F_\bullet </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> of the underlying functors that is compatible with the connecting homomorphisms, hence such that all these <a class="existingWikiWord" href="/nlab/show/commuting+square">squares commute</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><msub><mi>E</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>F</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>δ</mi> <mi>E</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>δ</mi> <mi>F</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover></mtd> <mtd><msub><mi>F</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>∅</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ E_{\bullet +1}(X,A) &\overset{\eta_{(X,A)}}{\longrightarrow}& F_{\bullet +1}(X,A) \\ {}^{\mathllap{\delta_E}}\downarrow && \downarrow^{\mathrlap{\delta_F}} \\ E_\bullet(A,\emptyset) &\overset{\eta_{(A,\emptyset)}}{\longrightarrow}& F_\bullet(A,\emptyset) } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="AlternativeFormulationOfExcisionAxiom"> <h6 id="lemma">Lemma</h6> <p>The excision axiom in def. <a class="maruku-ref" href="#GeneralizedHomologyTheory"></a> is equivalent to the following statement:</p> <p>For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A,B \hookrightarrow X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>A</mi><mo>∪</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">X = A \cup B</annotation></semantics></math>, then the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i \colon (A, A \cap B) \longrightarrow (X,B) </annotation></semantics></math></div> <p>induces an isomorphism,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> i_\ast \;\colon\; E_\bullet(A, A \cap B) \overset{\simeq}{\longrightarrow} E_\bullet(X, B) \,. </annotation></semantics></math></div></div> <p>(e.g <a href="#Switzer75">Switzer 75, 7.2, 7.5</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>First consider the statement under the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = Int(A) \cup Int(B)</annotation></semantics></math>.</p> <p>In one direction, suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> satisfies the original excision axiom. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A,B</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mo lspace="0em" rspace="thinmathspace">Int</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = \Int(A) \cup Int(B)</annotation></semantics></math>, set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>≔</mo><mi>X</mi><mo>−</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">U \coloneqq X-A</annotation></semantics></math> and observe that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mover><mi>U</mi><mo>¯</mo></mover></mtd> <mtd><mo>=</mo><mover><mrow><mi>X</mi><mo>−</mo><mi>A</mi></mrow><mo>¯</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>X</mi><mo>−</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⊂</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} \overline{U} & = \overline{X-A} \\ & = X- Int(A) \\ & \subset Int(B) \end{aligned} </annotation></semantics></math></div> <p>and that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>B</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X-U, B-U) = (A, A \cap B) \,. </annotation></semantics></math></div> <p>Hence the excision axiom implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo>∩</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X, B) \overset{\simeq}{\longrightarrow} E^\bullet(A, A \cap B)</annotation></semantics></math>.</p> <p>Conversely, suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup></mrow><annotation encoding="application/x-tex">E^\bullet</annotation></semantics></math> satisfies the alternative condition. Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↪</mo><mi>A</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \hookrightarrow A \hookrightarrow X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>U</mi><mo>¯</mo></mover><mo>⊂</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\overline{U} \subset Int(A)</annotation></semantics></math>, observe that we have a cover</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>Int</mi><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mover><mi>U</mi><mo>¯</mo></mover><mo stretchy="false">)</mo><mo>∩</mo><mo lspace="0em" rspace="thinmathspace">Int</mo><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>⊃</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∩</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Int(X-U) \cup Int(A) & = (X - \overline{U}) \cap \Int(A) \\ & \supset (X - Int(A)) \cap Int(A) \\ & = X \end{aligned} </annotation></semantics></math></div> <p>and that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>A</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X-U, (X-U) \cap A) = (X-U, A - U) \,. </annotation></semantics></math></div> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mi>A</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo>,</mo><mo stretchy="false">(</mo><mi>X</mi><mo>−</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>E</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> E^\bullet(X-U,A-U) \simeq E^\bullet(X-U, (X-U)\cap A) \simeq E^\bullet(X,A) \,. </annotation></semantics></math></div> <p>This shows the statement for the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∪</mo><mi>Int</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X = Int(A)\cup Int(U)</annotation></semantics></math>. The general statement reduces to this by finding a suitable homotopy equivalence to a slightly larger covering pair (e.g <a href="#Switzer75">Switzer 75, 7.5</a>).</p> </div> <div class="num_prop" id="ExactSequenceOfATriple"> <h6 id="proposition">Proposition</h6> <p><strong>(exact sequence of a triple)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">E_\bullet</annotation></semantics></math> an unreduced generalized cohomology theory, def. <a class="maruku-ref" href="#GeneralizedHomologyTheory"></a>, then every inclusion of two consecutive subspaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>↪</mo><mi>Y</mi><mo>↪</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> Z \hookrightarrow Y \hookrightarrow X </annotation></semantics></math></div> <p>induces a <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a> of homology groups of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>→</mo><msub><mi>E</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msub><mi>E</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><msub><mi>E</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover></mover><msub><mi>E</mi> <mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>⋯</mi></mrow><annotation encoding="application/x-tex"> \cdots \to E_q(Y,Z) \stackrel{}{\longrightarrow} E_q(X,Z) \stackrel{}{\longrightarrow} E_q(X,Y) \stackrel{\bar \delta}{\longrightarrow} E_{q-1}(Y,Z) \to \cdots </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><mover><mi>δ</mi><mo stretchy="false">¯</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>E</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>δ</mi></mover><msub><mi>E</mi> <mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>E</mi> <mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \bar \delta \;\colon \; E_{q}(X,Y) \stackrel{\delta}{\longrightarrow} E_{q-1}(Y) \longrightarrow E_{q-1}(Y,Z) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Apply the <a class="existingWikiWord" href="/nlab/show/braid+lemma">braid lemma</a> to the interlocking long exact sequences of the three pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Y)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,Z)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Y,Z)</annotation></semantics></math>:</p> <p><img src="http://www.ncatlab.org/nlab/files/BraidDiagramForHomologyOnTripled.jpg" width="500" /></p> <p>(graphics from <a href="http://math.stackexchange.com/a/1180681/58526">this Maths.SE comment</a>)</p> <p>See <a href="braid+lemma#ExactSequenceForTripleInGeneralizedHomology">here</a> for details.</p> </div> <h2 id="properties">Properties</h2> <h3 id="expression_by_ordinary_homology_via_atiyahhirzebruch_spectral_sequence">Expression by ordinary homology via Atiyah-Hirzebruch spectral sequence</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> serves to express generalized homology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">E_\bullet</annotation></semantics></math> in terms of ordinary homology with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">E_\bullet(\ast)</annotation></semantics></math>.</p> <h3 id="WhiteheadTheorem">Whitehead theorem</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>⟶</mo><mi>F</mi></mrow><annotation encoding="application/x-tex">\phi \colon E \longrightarrow F</annotation></semantics></math> be a morphism of reduced generalized (co-)homology functors, def. <a class="maruku-ref" href="#ReducedGeneralizedHomology"></a> (a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>) such that its component</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi(S^0) \colon E(S^0) \longrightarrow F(S^0) </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/0-sphere">0-sphere</a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(X)\colon E(X)\to F(X)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> for <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> any <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a> with a <a class="existingWikiWord" href="/nlab/show/finite+number">finite number</a> of cells. If both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <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> satisfy the <a class="existingWikiWord" href="/nlab/show/wedge+axiom">wedge axiom</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi(X)</annotation></semantics></math> is an isomorphism for <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> any <a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</a>, not necessarily finite.</p> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <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> <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a>/<a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> functors a proof of this is in (<a href="#EilenbergSteenrod52">Eilenberg-Steenrod 52, section III.10</a>). From this the general statement follows (e.g. <a href="#Kochman96">Kochman 96, theorem 3.4.3, corollary 4.2.8</a>) via the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of the <a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a> (the classical result gives that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> induces an isomorphism between the second pages of the AHSSs for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <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>). A complete proof of the general result is also given as (<a href="#Switzer75">Switzer 75, theorem 7.55, theorem 7.67</a>)</p> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+homotopy+homology+theory">stable homotopy homology theory</a> is the homology theory represented by the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+homology">ordinary homology</a> is the homology theory represented by an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+homology+theory">bordism homology theory</a> is the homology theory represented by a <a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a>;</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+cohomology">generalized cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kronecker+pairing">Kronecker pairing</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah-Hirzebruch+spectral+sequence">Atiyah-Hirzebruch spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phantom+map">phantom map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bivariant+cohomology">bivariant cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+equivalence">Bousfield equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+homomorphism">Hurewicz homomorphism</a>, <a class="existingWikiWord" href="/nlab/show/Boardman+homomorphism">Boardman homomorphism</a></p> </li> </ul> <h2 id="references">References</h2> <p>(For more see the references at <em><a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized (Eilenberg-Steenrod) cohomology</a></em>.)</p> <p>Original articles include</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/George+Whitehead">George Whitehead</a>, <em>Generalized homology theories</em> (1961) (<a href="http://www.maths.ed.ac.uk/~aar/papers/gww9.pdf">pdf</a>)</li> </ul> <p>Textbook accounts include</p> <ul> <li id="Kochmann96"> <p><a class="existingWikiWord" href="/nlab/show/Stanley+Kochmann">Stanley Kochmann</a>, section 3.4 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="Switzer75"> <p><a class="existingWikiWord" href="/nlab/show/Robert+Switzer">Robert Switzer</a>, chapter 7 (and 8-12) of <em>Algebraic Topology - Homotopy and Homology</em>, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, chapter II, section 6 of <em><a class="existingWikiWord" href="/nlab/show/Symmetric+spectra">Symmetric spectra</a></em>, 2012 (<a href="http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf">pdf</a>)</p> </li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Friedrich+Bauer">Friedrich Bauer</a>, <em>Classifying spectra for generalized homology theories</em> Annali di Maternatica pura ed applicata</p> <p>(IV), Vol. CLXIV (1993), pp. 365-399</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Friedrich+Bauer">Friedrich Bauer</a>, <em>Remarks on universal coefficient theorems for generalized homology theories</em> Quaestiones Mathematicae</p> <p>Volume 9, Issue 1 & 4, 1986, Pages 29 - 54</p> </li> </ul> <p>A general construction of homologies by “geometric cycles” similar to the <a class="existingWikiWord" href="/nlab/show/Baum-Douglas+geometric+cycles">Baum-Douglas geometric cycles</a> for <a class="existingWikiWord" href="/nlab/show/K-homology">K-homology</a> is discussed in</p> <ul> <li>S. Buoncristiano, C. P. Rourke and B. J. Sanderson, <em>A geometric approach to homology theory</em>, Cambridge Univ. Press, Cambridge, Mass. (1976)</li> </ul> <p>Further generalization of this to <a class="existingWikiWord" href="/nlab/show/bivariant+cohomology+theories">bivariant cohomology theories</a> is in</p> <ul> <li>Martin Jakob, <em>Bivariant theories for smooth manifolds</em>, Applied Categorical Structures 10 no. 3 (2002)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 4, 2023 at 05:18:54. 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