View source for Homology manifold

''generalized manifold''

A locally compact topological space whose local homological structure is analogous to the local structure of ordinary topological manifolds, including manifolds with boundary. More exactly, a homology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h0478002.png" />-manifold (a generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h0478004.png" />-manifold) over a group or a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h0478005.png" /> of coefficients is a locally compact topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h0478006.png" /> with finite homological dimension (cf. [[Homological dimension of a space|Homological dimension of a space]]) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h0478007.png" /> and such that all its local homology groups (cf. [[Local homology|Local homology]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h0478008.png" /> are trivial if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h0478009.png" />, and are isomorphic either to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780010.png" /> or zero if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780011.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780012.png" /> is the direct limit of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780013.png" />, taken over all neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780014.png" /> of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780016.png" /> is a [[Homology theory|homology theory]] that satisfies all the [[Steenrod–Eilenberg axioms|Steenrod–Eilenberg axioms]], including the exactness axiom. In the category of locally contractible spaces the theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780017.png" />, considered with compact support, is isomorphic to the singular theory (cf. [[Singular homology|Singular homology]]). The groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780018.png" /> automatically turn out to be the stalks of some sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780019.png" /> (cf. [[Sheaf theory|Sheaf theory]]), known as the orienting sheaf of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780020.png" />. A homology manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780021.png" /> is said to be orientable if the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780022.png" /> is isomorphic to the constant sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780023.png" />, and is said to be locally orientable if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780024.png" /> is locally constant at the points where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780025.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780026.png" /> is a [[Principal ideal ring|principal ideal ring]] and if all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780027.png" /> are non-zero, a homology manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780028.png" /> is always locally orientable. If a homology manifold over a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780029.png" /> is locally orientable, then the set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780030.png" /> on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780031.png" /> is closed, nowhere dense and forms the boundary of the homology manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780032.png" />. A locally orientable homology manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780033.png" /> has the same homological properties as ordinary manifolds.

E.g., the theorem on preservation of domain is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780035.png" />, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780036.png" /> is nowhere dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780037.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780038.png" />, etc.

For any homology manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780039.png" /> there are natural isomorphisms ([[Poincaré duality|Poincaré duality]])

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780040.png" /></td> </tr></table>

(cohomology with coefficients in a sheaf). Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780041.png" /> is any integer; however, the homological dimension of the homology manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780042.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780043.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780044.png" />, and thus the content of these isomorphisms is non-trivial only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780045.png" />. Similar isomorphisms are valid for homology and cohomology with support in any paracompactifying family (in particular, for homology and cohomology spaces with compact support). The condition of isomorphism between the non-zero stalks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780046.png" /> of the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780047.png" /> and the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780048.png" /> is immaterial. Instead of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780049.png" /> it is also possible to consider any locally constant sheaf of coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780050.png" /> with stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780051.png" /> (this is accompanied by a change in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780052.png" />). Any open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780053.png" /> is a homology manifold. For this reason, the use of the equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780054.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780055.png" /></td> </tr></table>

in the second one of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780056.png" /> has compact closure, while the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780057.png" /> indicates the compactness of the supports, makes it possible to obtain the isomorphisms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780058.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780059.png" /></td> </tr></table>

as special cases of Poincaré duality. Combination of the exact homology and cohomology sequences of the respective pairs also makes it possible to consider the isomorphisms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780060.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780061.png" /></td> </tr></table>

— the latter one being a generalization of [[Alexander duality|Alexander duality]] — as special cases of Poincaré duality. Similar relations are also valid for homology and cohomology with supports in a given fixed family.

Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780062.png" /></td> </tr></table>

let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780063.png" /> be compact and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780064.png" /> be a closed or an open subset. A consequence of the previous isomorphisms and of the exactness of the homology and cohomology is an isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780065.png" /></td> </tr></table>

which represents the [[Pontryagin duality|Pontryagin duality]] for a closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780066.png" /> and the [[Steenrod duality|Steenrod duality]] for an open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780067.png" />. This and the property of continuity of cohomology implies that the isomorphism

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780068.png" /></td> </tr></table>

is valid for any subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780069.png" /> (Sitnikov duality). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780070.png" /> is non-compact one must consider homology with supports closed in all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780071.png" /> rather than homology with compact supports. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780072.png" /> is compact, reduced homology must be used for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780073.png" />.

Non-trivial examples of homology manifolds include  "factors"  of ordinary manifolds such as Euclidean spaces: If for a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780074.png" /> there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780075.png" /> such that the Cartesian product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780076.png" /> is a homology manifold, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780078.png" /> are also homology manifolds. There are examples of homology manifolds that are not locally Euclidean at any one of their points. Homology manifolds play an important role in certain problems of transformation groups (cf. [[Transformation group|Transformation group]]), where they appear as [[Orbit|orbit]] spaces or as sets of fixed points.

There exists a cohomological variant of the definition of generalized manifolds. Any cohomology manifold over a principal ideal ring is a homology manifold over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780079.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780080.png" /> is at most countable, then the converse proposition is true as well.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Čech,   "Théorie générale des variétés et de leurs théorèmes de dualité"  ''Ann. of Math. (2)'' , '''34'''  (1933)  pp. 621–730</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Lefschetz,   "On generalized manifolds"  ''Amer. J. Math.'' , '''55'''  (1933)  pp. 469–504</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Aleksandroff,   "On local properties of closed sets"  ''Ann. of Math. (2)'' , '''36''' :  1  (1935)  pp. 1–35</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Aleksandroff,   L.S. [L.S. Pontryagin] Pontrjagin,   "Les variétés à <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047800/h04780081.png" /> dimensions généralisés"  ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''202'''  (1936)  pp. 1327–1329</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  P.A. Smith,   "Transformations of finite period"  ''Ann. of Math. (2)'' , '''40''' :  3  (1939)  pp. 690–711</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  E.G. Begle,   "Locally connected spaces and generalized manifolds"  ''Amer. J. Math.'' , '''64'''  (1942)  pp. 553–574</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  R. Wilder,   "Topology of manifolds" , Amer. Math. Soc.  (1949)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A. Borel,   "The Poincaré duality in generalized manifolds"  ''Michigan Math. J.'' , '''4'''  (1957)  pp. 227–239</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  C.T. Yang,   "Transformation groups on a homological manifold"  ''Trans. Amer. Math. Soc.'' , '''87'''  (1958)  pp. 261–283</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  P.E. Conner,   E.E. Floyd,   "A characterization of generalized manifolds"  ''Michigan Math. J.'' , '''6'''  (1959)  pp. 33–43</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  F. Raymond,   "Separation and union theorems for generalized manifolds with boundary"  ''Michigan Math. J.'' , '''7'''  (1960)  pp. 7–21</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  G.E. Bredon,   "Orientation in generalized manifolds and application to the theory of transformation groups"  ''Michigan Math. J.'' , '''7'''  (1960)  pp. 35–64</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  A. Borel,   "Homology and duality in generalized manifolds"  A. Borel (ed.) , ''Seminar on transformation groups'' , Princeton Univ. Press  (1960)  pp. 23–33</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  G.E. Bredon,   "Wilder manifolds are locally orientable"  ''Proc. Nat. Acad. Sci. USA'' , '''63''' :  4  (1969)  pp. 1079–1081</TD></TR></table>