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| In [[mathematics]], '''Borel−Moore homology''' or '''homology with closed support''' is a [[homology theory]] for [[locally compact space]]s, introduced by {{harvs|txt|last1=Borel|author1-link=Armand Borel|last2=Moore|author2-link=John Coleman Moore|year=1960}}.
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| For [[compact space]]s, the Borel−Moore homology coincide with the usual [[singular homology]], but for non-compact spaces, it usually gives homology groups with better properties.
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| Note: There is an equivariant [[cohomology]] theory for spaces upon which a group <math>G</math> acts which is also called Borel cohomology and is defined as <math>H^*_G(X) = H^*(EG\times_G X)</math>. This is not related to the subject of this article.
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| ==Definition==
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| There are several ways to define Borel−Moore homology. They all coincide for spaces <math> \ X </math> that are homotopy equivalent to a finite [[CW complex]] and admit a closed embedding into a smooth manifold <math> \ M </math> such that <math>\ X </math> is a [[retract]] of an open neighborhood of itself in <math>\ M </math>.
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| ===Definition via locally finite chains===
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| Let <math>\ T </math> be a [[Simplicial complex|triangulation]] of <math>\ X </math>. Denote by <math>\ C_i ^T ((X)) </math> the vector space of formal (infinite) sums
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| :<math> \xi = \sum _{\sigma \in T^{(i)} } \xi _{\sigma } \sigma. </math>
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| Note that for each element
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| :<math>\ \xi \in C_i ^T((X)) , </math>
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| its [[support (mathematics)|support]],
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| :<math>\ |\xi | = \bigcup _{\xi _{\sigma}\neq 0}\sigma, </math>
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| is closed. The support is compact if and only if <math>\ \xi </math> is a finite linear combination of simplices.
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| The space
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| :<math> \ C_i ((X)) </math>
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| of i-chains with closed support is defined to be the [[direct limit]] of | |
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| :<math>\ C_i ^T ((X)) </math>
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| under refinements of <math>\ T </math>. The boundary map of simplicial homology extends to a boundary map
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| :<math>\ \partial :C_i((X))\to C_{i-1}((X)) </math>
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| and it is easy to see that the sequence
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| :<math> \dots \to C_{i+1} ((X)) \to C_i ((X)) \to C_{i-1} ((X)) \to \dots </math>
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| is a [[chain complex]]. The '''Borel−Moore homology''' of ''X'' is defined to be the homology of this chain complex. Concretely,
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| :<math> H^{BM} _i (X) =Ker (\partial :C_i ((X)) \to C_{i-1} ((X)) )/ Im (\partial :C_{i+1} ((X)) \to C_i ((X)) ). </math>
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| ===Definition via compactifications===
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| Let <math>\ \bar{X} </math> be a [[compactification (mathematics)|compactification]] of <math>\ X </math> such that the pair
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| <math>\ (\bar{X} , \bar{X} \setminus X) </math>
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| is a [[CW-pair]]. For example, one may take the [[one point compactification]] of <math>\ X </math>. Then
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| :<math> \ H^{BM}_i(X)=H_i(\bar{X} , \bar{X} \setminus X), </math>
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| where in the right hand side, usual [[relative homology]] is meant.
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| ===Definition via Poincaré duality===
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| Let <math>\ X \subset M </math> be a closed embedding of <math>\ X </math> in a [[smooth manifold]] of dimension ''m'', such that <math>\ X </math> is a retract of an open neighborhood of itself. Then
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| :<math>\ H^{BM}_i(X)= H^{m-i}(M,M\setminus X),</math>
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| where in the right hand side, usual relative cohomology is meant.
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| ===Definition via the dualizing complex===
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| Let
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| <math>\ \mathbb{D} _X </math>
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| be the [[Verdier duality|dualizing complex]] of <math>\ \ X </math>. Then
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| :<math>\ H^{BM}_i (X)=H^{-i} (X,\mathbb{D} _X), </math>
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| where in the right hand side, [[hypercohomology]] is meant.
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| ==Properties==
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| * Borel−Moore homology is '''not''' [[homotopy invariant]]. For example,
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| :<math>\ H^{BM}_i(\mathbb{R} ^n ) </math> | |
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| vanishes for <math>\ i\neq n </math> and equals <math>\ \mathbb{R} </math> for <math>\ i=n </math>.
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| * Borel−Moore homology is a [[covariant functor]] with respect to [[proper map]]s. Suppose <math>\ f:X\to Y </math> is a proper map. Then <math>\ f </math> induces a continuous map <math>\ \bar{f} :(\bar{X} , \bar{X} \setminus X )\to (\bar {Y} , \bar{Y} \setminus Y) </math> where <math> \bar{X}=X\cup \{ \infty \} , \bar{Y}=Y\cup \{ \infty \} </math> are the one point compactifications. Using the definition of Borel−Moore homology via compactification, there is a map <math>\ f_*:H^{BM}_* (X)\to H^{BM}_* (Y) </math>. Properness is essential, as it guarantees that the induced map on compactifications will be continuous. There is no pushforward for a general continuous map of spaces. As a counterexample, one can consider the non-proper inclusion <math> \mathbb{C}^* \to \mathbb{C}</math> .
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| * If <math>\ F \subset X </math> is a closed set and <math>\ U=X\setminus F </math> is its complement, then there is a long exact sequence
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| <math> \dots \to H^{BM}_i (F) \to H^{BM}_i (X) \to H^{BM}_i (U) \to H^{BM}_{i-1} (F) \to \dots </math> .
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| * One of the main reasons to use Borel−Moore homology is that for every [[orientable manifold]] (in particular, for every smooth complex variety) <math>\ M </math>, there is a [[fundamental class]] <math>\ [M]\in H^{BM}_{top}(M) </math>. This is just the sum over all top dimensional simplices in a specific triangulation. In fact, in Borel−Moore homology, one can define a fundamental class for arbitrary (i.e. possibly singular) complex varieties. In this case the set of smooth points <math>\ M^{reg} \subset M </math> has complement of (real) [[codimension]] 2 and by the long exact sequence above the top dimensional homologies of <math> \ M </math> and <math> \ M^{reg} </math> are canonically isomorphic. One then defines the fundamental class of <math>\ M </math> to be the fundamental class of <math>\ M^{reg} </math>.
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| ==References==
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| *Iversen, Birger ''Cohomology of sheaves.'' Universitext. Springer-Verlag, Berlin, 1986. xii+464 pp. ISBN 3-540-16389-1 {{MR|0842190}}
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| *{{Citation | last1=Borel | first1=Armand | author1-link=Armand Borel | last2=Moore | first2=John C. | title=Homology theory for locally compact spaces | url=http://projecteuclid.org/euclid.mmj/1028998385 | mr=0131271 | year=1960 | journal=The Michigan Mathematical Journal | issn=0026-2285 | volume=7 | pages=137–159}}
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| {{DEFAULTSORT:Borel-Moore homology}}
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| [[Category:Homology theory]]
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| [[Category:Sheaf theory]]
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