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| | Hello, my title is Andrew and my spouse doesn't like it at all. My spouse and I live in Mississippi and I adore each working day living here. Distributing production is how he makes a residing. My spouse doesn't like it the way I do but what I really like performing is caving but I don't have the time lately.<br><br>my site: [https://Www-ocl.Gist.ac.kr/work/xe/?document_srl=605236 certified psychics] |
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| In [[algebraic topology]] the '''cap product''' is a method of adjoining a [[chain (algebraic topology)|chain]] of degree ''p'' with a [[cochain]] of degree ''q'', such that ''q'' ≤ ''p'', to form a composite chain of degree ''p'' − ''q''. It was introduced by [[Eduard Čech]] in 1936, and independently by [[Hassler Whitney]] in 1938.
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| ==Definition==
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| Let ''X'' be a [[topological space]] and ''R'' a coefficient ring. The cap product is a [[bilinear map]] on singular homology and cohomology
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| :<math>\frown\ : H_p(X;R)\times H^q(X;R) \rightarrow H_{p-q}(X;R).</math>
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| defined by contracting a [[singular chain]] <math>\sigma : \Delta\ ^p \rightarrow\ X</math> with a singular [[cochain]] <math> \psi \in C^q(X;R), </math> by the formula :
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| :<math> \sigma \frown \psi = \psi(\sigma|_{[v_0, \ldots, v_q]}) \sigma|_{[v_q, \ldots, v_p]}.</math>
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| Here, the notation <math>\sigma|_{[v_0, \ldots, v_q]}</math> indicates the restriction of the simplicial map <math>\sigma</math> to its face spanned by the vectors of the base, see [[Simplex]].
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| ==Interpretation==
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| In analogy with the interpretation of the [[cup product]] in terms of the [[Künneth formula]], we can explain the existence of the cap product by considering the composition
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| <math> C_\bullet(X) \otimes C^\bullet(X) \overset{\Delta_* \otimes \mathrm{Id}}{\longrightarrow} C_\bullet(X) \otimes C_\bullet(X) \otimes C^\bullet(X) \overset{\mathrm{Id} \otimes \varepsilon}{\longrightarrow} C_\bullet(X) </math>
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| in terms of the [[chain complex|chain]] and cochain complexes of <math>X</math>, where we are taking [[Künneth theorem|tensor products of chain complexes]], <math> \Delta \colon X \to X \times X</math> is the [[diagonal functor|diagonal map]] which induces the map <math>\Delta_*</math> on the chain complex, and <math>\varepsilon \colon C_p(X) \otimes C^q(X) \to \mathbb{Z}</math> is the [[evaluation map]] (always 0 except for <math>p=q</math>).
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| This composition then passes to the quotient to define the cap product <math> \frown \colon H_\bullet(X) \times H^\bullet(X) \to H_\bullet(X)</math>, and looking carefully at the above composition shows that it indeed takes the form of maps <math> \frown \colon H_p(X) \times H^q(X) \to H_{p-q}(X)</math>, which is always zero for <math>p < q</math>.
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| ==The slant product==
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| The above discussion indicates that the same operation can be defined on [[cartesian product]]s <math>X\times Y</math> yielding a product
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| :<math>\backslash\ : H_p(X;R)\otimes H^q(X\times Y;R) \rightarrow H^{q-p}(Y;R).</math>
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| In case ''X = Y'', the two products are related by the diagonal map.
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| ==Equations==
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| The boundary of a cap product is given by :
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| :<math>\partial(\sigma \frown \psi) = (-1)^q(\partial \sigma \frown \psi - \sigma \frown \delta \psi). </math>
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| Given a map ''f'' the induced maps satisfy :
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| :<math> f_*( \sigma ) \frown \psi = f_*(\sigma \frown f^* (\psi)). </math>
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| The cap and [[cup product]] are related by :
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| :<math> \psi(\sigma \frown \varphi) = (\varphi \smile \psi)(\sigma)</math>
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| where
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| :<math>\sigma : \Delta ^{p+q} \rightarrow X</math> , <math> \psi \in C^q(X;R)</math>and <math> \varphi \in C^p(X;R). </math>
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| An interesting consequence of the last equation is that it makes <math>H_{\ast}(X;R)</math> into a right <math>H^{\ast}(X;R)-</math> [[module (mathematics)|module]].
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| ==See also==
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| *[[cup product]]
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| *[[Poincaré duality]]
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| *[[singular homology]]
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| *[[homology theory]]
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| ==References==
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| *[[Allen Hatcher|Hatcher, A.]], ''[http://www.math.cornell.edu/~hatcher/AT/ATchapters.html Algebraic Topology],'' [[Cambridge University Press]] (2002) ISBN 0-521-79540-0. Detailed discussion of homology theories for simplicial complexes and manifolds, singular homology, etc.
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| *{{nlab|id=slant+product|title=slant product}}
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| {{DEFAULTSORT:Cap Product}}
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| [[Category:Homology theory]]
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| [[Category:Algebraic topology]]
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| [[Category:Binary operations]]
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Hello, my title is Andrew and my spouse doesn't like it at all. My spouse and I live in Mississippi and I adore each working day living here. Distributing production is how he makes a residing. My spouse doesn't like it the way I do but what I really like performing is caving but I don't have the time lately.
my site: certified psychics