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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='#DefinitionInTermsOfExtendedSquares'>Construction in terms of extended squares</a></li> <li><a href='#axiomatic_characterization'>Axiomatic characterization</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#relation_to_bockstein_homomorphism'>Relation to Bockstein homomorphism</a></li> <li><a href='#compatibility_with_suspension'>Compatibility with suspension</a></li> <li><a href='#relation_to_massey_products'>Relation to Massey products</a></li> <li><a href='#adem_relations'>Adem relations</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#HopfInvariant'>Hopf invariant</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>, what are called the <em>Steenrod squares</em> is the system of <a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a> on <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+2">cyclic group of order 2</a>) which is compatible with <a class="existingWikiWord" href="/nlab/show/suspension">suspension</a> (the “stable cohomology operations”). They are special examples of <a class="existingWikiWord" href="/nlab/show/power+operations">power operations</a>.</p> <p>The collection of Steenrod squares for all degrees forms the <em><a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a></em>, see there for more.</p> <h2 id="definition">Definition</h2> <h3 id="DefinitionInTermsOfExtendedSquares">Construction in terms of extended squares</h3> <p>We discuss the explicit construction of the Steenrod-operations in terms of <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</a> of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> equipped with a suitable product. We follow (<a href="#Lurie07">Lurie 07, lecture 2</a>).</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub><mo>≔</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/field">field</a> with two elements.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>, hence an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, and 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>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>V</mi> <mrow><mi>h</mi><msub><mi>Σ</mi> <mi>n</mi></msub></mrow> <mrow><mo>⊗</mo><mi>n</mi></mrow></msubsup><mo>∈</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> with itself by the <a class="existingWikiWord" href="/nlab/show/action">action</a> of the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>. Explicitly this is presented, up to <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> by the ordinary <a class="existingWikiWord" href="/nlab/show/coinvariants">coinvariants</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D_n(V)</annotation></semantics></math> of the tensor product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">V^{\otimes n}</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/free+resolution">free resolution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><msubsup><mi>Σ</mi> <mi>n</mi> <mo>•</mo></msubsup></mrow><annotation encoding="application/x-tex">E \Sigma_n^\bullet</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_2</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>V</mi> <mrow><mi>h</mi><msub><mi>Σ</mi> <mi>n</mi></msub></mrow> <mrow><mo>⊗</mo><mi>n</mi></mrow></msubsup><mo>≃</mo><msub><mi>D</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msup><mi>V</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>⊗</mo><mi>E</mi><msub><mi>Σ</mi> <mi>n</mi></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>Σ</mi> <mi>n</mi></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V^{\otimes n}_{h \Sigma_n} \simeq D_n(V) \coloneqq (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n} \,. </annotation></semantics></math></div> <p>This is called the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <em><a class="existingWikiWord" href="/nlab/show/extended+power">extended power</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <p>For instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mi>n</mi><mo stretchy="false">]</mo><mo>⊗</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><msub><mi>Σ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> D_2( \mathbb{F}_2[-n]) \simeq \mathbb{F}_2[-2n] \otimes C^\bullet(B \Sigma_2) \,, </annotation></semantics></math></div> <p>where on the right we have the, say, <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complex</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>Σ</mi> <mn>2</mn></msub><mo>≃</mo><mi>B</mi><msub><mi>Σ</mi> <mn>2</mn></msub><mo>≃</mo><mi>ℝ</mi><msup><mi>P</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">\ast //\Sigma_2 \simeq B \Sigma_2 \simeq \mathbb{R}P^\infty</annotation></semantics></math>, which is the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> D_2(V) \longrightarrow V </annotation></semantics></math></div> <p>is called a <em>symmetric multiplication</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> (a shadow of an <a class="existingWikiWord" href="/nlab/show/E-infinity+algebra">E-infinity algebra</a> structure). The archetypical class of examples of these are given by the <a class="existingWikiWord" href="/nlab/show/singular+cohomology">singular cohomology</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V = C^\bullet(X, \mathbb{F}_2)</annotation></semantics></math> of any <a class="existingWikiWord" href="/nlab/show/topological+space">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>, for instance of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">B \Sigma_2</annotation></semantics></math>.</p> <p>Therefore there is a canonical <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</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> <mi>k</mi></msup><mo stretchy="false">(</mo><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>H</mi> <mrow><mn>2</mn><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub><mo stretchy="false">(</mo><mi>B</mi><msub><mi>Σ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><msub><mi>e</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> H^k(D_2(\mathbb{F}_2[-n])) \simeq H_{2n - k}(B \Sigma_2, \mathbb{F}_2) e_{2n} </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> of the extended square of the chain compplex concentrated on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_2</annotation></semantics></math> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a> of this classifying space shifted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi></mrow><annotation encoding="application/x-tex">2 n</annotation></semantics></math>.</p> <p>Using this one gets for general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> and for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">i \leq n</annotation></semantics></math> a map that sends an element in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a></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>v</mi><mo stretchy="false">]</mo><mo>∈</mo><msup><mi>H</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> [v] \in H^n(V) </annotation></semantics></math></div> <p>represented by a morphism of <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>v</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> v \;\colon\; \mathbb{F}_2[-n] \longrightarrow V </annotation></semantics></math></div> <p>to the element</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mover><mi>Sq</mi><mo>¯</mo></mover> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \overline{Sq}^i(v) \in H^{n+1}(D_2(V)) </annotation></semantics></math></div> <p>represented by the <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</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>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo>−</mo><mi>i</mi><mo stretchy="false">]</mo><mover><mo>⟶</mo><mn>1</mn></mover><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><msub><mi>Σ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mover><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{F}_2[-n-i] \stackrel{1}{\longrightarrow} C^\bullet(B \Sigma_2, \mathbb{F}_2) \stackrel{\simeq}{\longrightarrow} D_2(\mathbb{F}_2[-n]) \stackrel{D_2(v)}{\longrightarrow} D_2(V) \,. </annotation></semantics></math></div> <p>If moreover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is equipped with a <em>symmetric product</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">D_2(V) \longrightarrow V</annotation></semantics></math> as above, then one can further compose and form the element</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> {Sq}^i(v) \in H^{n+1}(V) </annotation></semantics></math></div> <p>represented by the <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</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>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo>−</mo><mi>i</mi><mo stretchy="false">]</mo><mover><mo>⟶</mo><mn>1</mn></mover><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>B</mi><msub><mi>Σ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow></mover><msub><mi>D</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>V</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{F}_2[-n-i] \stackrel{1}{\longrightarrow} C^\bullet(B \Sigma_2, \mathbb{F}_2) \stackrel{\simeq}{\longrightarrow} D_2(\mathbb{F}_2[-n]) \stackrel{D_2(v)}{\longrightarrow} D_2(V) \longrightarrow V \,. </annotation></semantics></math></div> <p>This <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>i</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sq^i \;\colon\; H^\bullet(V) \longrightarrow H^{\bullet + i}(V) </annotation></semantics></math></div> <p>is called the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th <em>Steenrod operation</em> or the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>th <em>Steenrod square</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>. By default this is understood for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V = C^\bullet(X,\mathbb{F}_2)</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/singular+homology">singular cochain complex</a> of some <a class="existingWikiWord" href="/nlab/show/topological+space">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>, as in the above examples, in which case it has the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>i</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>i</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sq^i \;\colon\; H^\bullet(X, \mathbb{F}_2) \longrightarrow H^{\bullet+i}(X,\mathbb{F}_2) \,. </annotation></semantics></math></div> <h3 id="axiomatic_characterization">Axiomatic characterization</h3> <p>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> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mi>n</mi></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">B^n \mathbb{Z}_2</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> of <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in the <a class="existingWikiWord" href="/nlab/show/group+of+order+2">group of order 2</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math> (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><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbb{Z}_2,n)</annotation></semantics></math>), regarded as an <a class="existingWikiWord" href="/nlab/show/object">object</a> in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> <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">H</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">of topological spaces</a>).</p> <p>Notice that 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 topological space (<a class="existingWikiWord" href="/nlab/show/CW-complex">CW-complex</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> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>≔</mo><mi>H</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>B</mi> <mi>n</mi></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^n(X, \mathbb{Z}_2) \coloneqq H(X, B^n \mathbb{Z}_2) </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of <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> in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>. Therefore, by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</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> <mi>k</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mi>l</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{k}(-, \mathbb{Z}_2) \to H^l(-, \mathbb{Z}_2) </annotation></semantics></math></div> <p>correspond bijectively to morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mi>k</mi></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><msup><mi>B</mi> <mi>l</mi></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">B^k \mathbb{Z}_2 \to B^l \mathbb{Z}_2</annotation></semantics></math>.</p> <p>The following characterization is due to (<a href="#SteenrodEpstein">SteenrodEpstein</a>).</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>The <strong>Steenrod squares</strong> are a collection of <a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Sq^n \;\colon\; H^k(-, \mathbb{Z}_2) \longrightarrow H^{k+n}(-, \mathbb{Z}_2) \,, </annotation></semantics></math></div> <p>hence of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>B</mi> <mi>k</mi></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⟶</mo><msup><mi>B</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi></mrow></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> Sq^n \;\colon\; B^k \mathbb{Z}_2 \longrightarrow B^{k + n} \mathbb{Z}_2 </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>,</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n,k \in \mathbb{N}</annotation></semantics></math> satisfying the following conditions:</p> <ol> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> it is the <a class="existingWikiWord" href="/nlab/show/identity">identity</a>;</p> </li> <li> <p>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mi>deg</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n \gt deg(x)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">Sq^n(x) = 0</annotation></semantics></math>;</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k = n</annotation></semantics></math> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>n</mi></msup><mo>:</mo><msup><mi>B</mi> <mi>n</mi></msup><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><msup><mi>B</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">Sq^n : B^n \mathbb{Z}_2 \to B^{2n} \mathbb{Z}_2</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cup+product">cup product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><mi>x</mi><mo>∪</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x \mapsto x \cup x</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo>∪</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>=</mo><mi>n</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>Sq</mi> <mi>i</mi></msup><mi>x</mi><mo stretchy="false">)</mo><mo>∪</mo><mo stretchy="false">(</mo><msup><mi>Sq</mi> <mi>j</mi></msup><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sq^n(x \cup y) = \sum_{i + j = n} (Sq^i x) \cup (Sq^j y)</annotation></semantics></math>;</p> </li> </ol> </div> <p>An analogous definition works for <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> for any <a class="existingWikiWord" href="/nlab/show/prime+number">prime number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>></mo><mn>2</mn></mrow><annotation encoding="application/x-tex">p \gt 2</annotation></semantics></math>. The corresponding operations are then usually denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>B</mi> <mi>k</mi></msup><msub><mi>ℤ</mi> <mi>p</mi></msub><mo>⟶</mo><msup><mi>B</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi></mrow></msup><msub><mi>ℤ</mi> <mi>p</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P^n \;\colon\; B^k \mathbb{Z}_p \longrightarrow B^{k+n} \mathbb{Z}_{p} \,. </annotation></semantics></math></div> <p>Under <a class="existingWikiWord" href="/nlab/show/composition">composition</a>, the Steenrod squares form an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_2</annotation></semantics></math>, called the <em><a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a></em>. See there for more.</p> <h2 id="properties">Properties</h2> <h3 id="relation_to_bockstein_homomorphism">Relation to Bockstein homomorphism</h3> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">Sq^1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a> of the <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>ℤ</mi> <mn>4</mn></msub><mo>→</mo><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \to \mathbb{Z}_4 \to \mathbb{Z}_2</annotation></semantics></math>.</p> <h3 id="compatibility_with_suspension">Compatibility with suspension</h3> <p>The Steenrod squares are compatible with the <a class="existingWikiWord" href="/nlab/show/suspension+isomorphism">suspension isomorphism</a>.</p> <p>Therefore the Steenrod squares are often also referred to as the <em><a class="existingWikiWord" href="/nlab/show/stabilization">stable</a> <a class="existingWikiWord" href="/nlab/show/cohomology+operations">cohomology operations</a></em></p> <h3 id="relation_to_massey_products">Relation to Massey products</h3> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Massey+product">Massey product</a></em>, <em><a href="Massey+product#RelationToSteenrodSquares">Relation to Steenrod squares</a></em></p> <h3 id="adem_relations">Adem relations</h3> <div class="num_prop" id="AdemRelations"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Adem+relations">Adem relations</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of Steenrod square operations satisfies the following relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>i</mi></msup><mo>∘</mo><msup><mi>Sq</mi> <mi>j</mi></msup><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>i</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></munder><msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mrow><mi>j</mi><mo>−</mo><mi>k</mi><mo>−</mo><mn>1</mn></mrow></mrow><mrow><mrow><mi>i</mi><mo>−</mo><mn>2</mn><mi>k</mi></mrow></mrow></mfrac><mo>)</mo></mrow> <mrow><mi>mod</mi><mn>2</mn></mrow></msub><msup><mi>Sq</mi> <mrow><mi>i</mi><mo>+</mo><mi>j</mi><mo>−</mo><mi>k</mi></mrow></msup><mo>∘</mo><msup><mi>Sq</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex"> Sq^i \circ Sq^j = \sum_{0 \leq k \leq i/2} \left( { { j - k - 1 } \atop { i - 2k } } \right)_{mod 2} Sq^{i + j -k} \circ Sq^k </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mi>i</mi><mo><</mo><mn>2</mn><mi>j</mi></mrow><annotation encoding="application/x-tex">0 \lt i \lt 2 j</annotation></semantics></math>.</p> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mi>a</mi></mrow><mrow><mi>b</mi></mrow></mfrac><mo>)</mo></mrow><mo>≔</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\left( a \atop b \right) \coloneqq 0</annotation></semantics></math> if <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 \lt b</annotation></semantics></math>.</p> </div> <div class="num_example" id="CompositionWithSq1"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Adem+relation">Adem relation</a> for <a class="existingWikiWord" href="/nlab/show/postcomposition">postcomposition</a> with the <a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mn>1</mn></msup><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">Sq^1 = \beta</annotation></semantics></math>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">j \geq 2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">i =1</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Adem+relations">Adem relations</a> (prop. <a class="maruku-ref" href="#AdemRelations"></a>) say 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><msup><mi>Sq</mi> <mn>1</mn></msup><mo>∘</mo><msup><mi>Sq</mi> <mi>j</mi></msup></mtd> <mtd><mo>=</mo><munder><munder><mrow><msub><mrow><mo>(</mo><mfrac linethickness="0"><mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow></mrow><mrow><mn>1</mn></mrow></mfrac><mo>)</mo></mrow> <mrow><mi>mod</mi><mn>2</mn></mrow></msub></mrow><mo>⏟</mo></munder><mrow><mo stretchy="false">(</mo><mi>j</mi><mo>−</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mrow><mi>mod</mi><mn>2</mn></mrow></msub></mrow></munder><msup><mi>Sq</mi> <mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msup><mi>Sq</mi> <mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>j</mi><mspace width="thinmathspace"></mspace><mtext>even</mtext></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo></mtd> <mtd><mi>j</mi><mspace width="thinmathspace"></mspace><mtext>odd</mtext></mtd></mtr></mtable></mrow></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Sq^1 \circ Sq^j & = \underset{ (j-1)_{mod 2} }{ \underbrace{ \left( { {j - 1 } \atop 1 } \right)_{mod 2} }} Sq^{j + 1} \\ & = \left\{ \array{ Sq^{j+1} &\vert& j \, \text{even} \\ 0 &\vert& j \, \text{odd} } \right. \end{aligned} </annotation></semantics></math></div></div> <p>This gives rise to:</p> <div class="num_example" id="IntegralSteenrodSquares"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/integral+Steenrod+squares">integral Steenrod squares</a>)</strong></p> <p>For <a class="existingWikiWord" href="/nlab/show/odd+natural+numbers">odd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">2n + 1 \in \mathbb{N}</annotation></semantics></math> defines the <a class="existingWikiWord" href="/nlab/show/integral+Steenrod+squares">integral Steenrod squares</a> to be</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Sq</mi> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>β</mi><mo>∘</mo><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sq^{2n + 1}_{\mathbb{Z}} \;\coloneqq\; \beta \circ Sq^{2n} \,. </annotation></semantics></math></div> <p>By example <a class="maruku-ref" href="#CompositionWithSq1"></a> and by <a href="Bockstein+homomorphism#Mod2BocksteinIntoMod2Cohomology">this example</a> these indeed are lifts of the odd <a class="existingWikiWord" href="/nlab/show/Steenrod+squares">Steenrod squares</a>:</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>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo stretchy="false">)</mo><mo>∘</mo><msubsup><mi>Sq</mi> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (mod\, 2) \circ Sq^{2n + 1}_{\mathbb{Z}} \;=\; Sq^{2n+1} \,, </annotation></semantics></math></div> <p>in that we have</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><msubsup><mi>Sq</mi> <mi>ℤ</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msubsup></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msup><mi>B</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></mover></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mi>β</mi></mover></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>ℤ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>id</mi></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>B</mi> <mrow><mi>k</mi><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>mod</mi><mspace width="thinmathspace"></mspace><mn>2</mn><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msup><mi>B</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msup><mi>Sq</mi> <mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></munder></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msup><mi>Sq</mi> <mn>1</mn></msup></mrow></munder></mtd> <mtd><msup><mi>B</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Sq^{2n+1}_{\mathbb{Z}} &\colon& B^{\bullet} (\mathbb{Z}/2\mathbb{Z}) &\overset{Sq^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\overset{ \beta }{\longrightarrow}& B^{\bullet + 2n + 1} \mathbb{Z} \\ && \downarrow^{ id } && \downarrow^{ id } && \downarrow^{\mathrlap{B^{k + 2 n + 1}(mod\, 2)}} \\ Sq^{2n+1} &\colon& B^{\bullet} (\mathbb{Z}/2\mathbb{Z}) &\underset{Sq^{2n}}{\longrightarrow}& B^{\bullet + 2n} (\mathbb{Z}/2\mathbb{Z}) &\underset{ Sq^1 }{\longrightarrow}& B^{\bullet + 2n + 1} (\mathbb{Z}/2\mathbb{Z}) } </annotation></semantics></math></div></div> <h2 id="examples">Examples</h2> <h3 id="HopfInvariant">Hopf invariant</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>S</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>→</mo><msup><mi>S</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\phi \colon S^{k+n-1} \to S^k</annotation></semantics></math>, a map of <a class="existingWikiWord" href="/nlab/show/spheres">spheres</a>, the Steenrod square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Sq</mi> <mi>n</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>H</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><msup><mi>H</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>cofib</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>𝔽</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sq^n \colon H^k(cofib(\phi), \mathbb{F}_2) \longrightarrow H^{k+n}(cofib(\phi),\mathbb{F}_2) </annotation></semantics></math></div> <p>(on the <a class="existingWikiWord" href="/nlab/show/homotopy+cofiber">homotopy cofiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cofib</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>S</mi> <mi>k</mi></msup><munder><mo>∪</mo><mrow><msup><mi>S</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></munder><msup><mi>D</mi> <mrow><mi>k</mi><mo>+</mo><mi>n</mi></mrow></msup></mrow><annotation encoding="application/x-tex">cofib(\phi)\simeq S^k \underset{S^{k+n-1}}{\cup} D^{k+n}</annotation></semantics></math>)</p> <p>is non-vanishing exactly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>8</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">n \in \{1,2,4,8\}</annotation></semantics></math>.</p> </div> <p>(<a href="Hopf+invariant+one#Adams60">Adams 60, theorem 1.1.1</a>).</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Hopf+invariant+one+theorem">Hopf invariant one theorem</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cup-i+product">cup-i product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology+operation">cohomology operation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bockstein+homomorphism">Bockstein homomorphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wu+class">Wu class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a></p> </li> </ul> <h2 id="references">References</h2> <p>The operations were first defined in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, <em>Products of cocycles and extensions of mappings</em>, Annals of Mathematics Second Series, Vol. 48, No. 2 (Apr., 1947), pp. 290-320 (<a href="https://www.jstor.org/stable/1969172">jstor:1969172</a>)</li> </ul> <p>The axiomatic definition appears in</p> <ul> <li id="SteenrodEpstein"><a class="existingWikiWord" href="/nlab/show/Norman+Steenrod">Norman Steenrod</a>, <a class="existingWikiWord" href="/nlab/show/David+Epstein">David Epstein</a>, <em>Cohomology operations</em>, Annals of Mathematics Studies, Princeton University Press (1962) (<a href="https://www.jstor.org/stable/j.ctt1b7x52h">jstor:j.ctt1b7x52h</a>)</li> </ul> <p>Lecture notes on Steenrod squares and the <a class="existingWikiWord" href="/nlab/show/Steenrod+algebra">Steenrod algebra</a> include</p> <ul> <li id="Lurie07"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, 18.917 <em><a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007">Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007</a></em>. (MIT OpenCourseWare: Massachusetts Institute of Technology), <em><a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/">Lecture notes</a></em></p> <p>Lecture 2 <em>Steenrod operations</em> (<a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture2.pdf">pdf</a>)</p> <p>Lecture 3 <em>Basic properties of Steenrod operations</em> (<a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture3.pdf">pdf</a>)</p> <p>Lecture 4 <em>The Adem relations</em> (<a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture4.pdf">pdf</a>)</p> <p>Lecture 5 <em>The Adem relations (cont.)</em> (<a href="http://ocw.mit.edu/courses/mathematics/18-917-topics-in-algebraic-topology-the-sullivan-conjecture-fall-2007/lecture-notes/lecture5.pdf">pdf</a>)</p> </li> </ul> <p>See also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wen-Tsun+Wu">Wen-Tsun Wu</a>, <em>Sur les puissances de Steenrod</em>, Colloque de Topologie de Strasbourg, IX, La Bibliothèque Nationale et Universitaire de Strasbourg, (1952)</p> </li> <li id="GonzalezDiazReal99"> <p><a class="existingWikiWord" href="/nlab/show/Rocio+Gonzalez-Diaz">Rocio Gonzalez-Diaz</a>, <a class="existingWikiWord" href="/nlab/show/Pedro+Real">Pedro Real</a>, <em>A Combinatorial Method for Computing Steenrod Squares</em>, Journal of Pure and Applied Algebra 139 (1999) 89-108 (<a href="https://arxiv.org/abs/math/0110308">arXiv:math/0110308</a>)</p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/David+W%C3%A4rn">David Wärn</a>, Section 4.4 of: <em>Eilenberg-MacLane spaces and stabilisation in homotopy type theory</em> [<a href="https://arxiv.org/abs/2301.03685">arXiv:2301.03685</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 17, 2023 at 05:54:31. 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