|
|
| Line 1: |
Line 1: |
| In [[mathematics]], a '''homology manifold''' (or '''generalized manifold''')
| | Hello. Let me introduce the author. Her name is Emilia Shroyer but it's not the most female title out there. To play baseball is the hobby he will never stop doing. Hiring is my profession. South Dakota is her beginning place but she needs to transfer simply because of her family.<br><br>My weblog - [http://www.1a-pornotube.com/user/WCrompton at home std testing] |
| is a [[locally compact topological space]] ''X'' that looks locally like a [[topological manifold]] from the point of view of [[homology theory]].
| |
| | |
| ==Definition==
| |
| A '''homology ''G''-manifold''' (without boundary) of dimension ''n'' over an abelian group ''G'' of coefficients is a locally compact topological space X with finite ''G''-[[cohomological dimension]] such that for any ''x''∈''X'', the homology groups
| |
| :<math> H_p(X,X-x, G)</math>
| |
| are trivial unless ''p''=''n'', in which case they are isomorphic to ''G''. Here ''H'' is some homology theory, usually singular homology. Homology manifolds are the same as homology '''Z'''-manifolds.
| |
| | |
| More generally, one can define homology manifolds with boundary, by allowing the local homology groups to vanish
| |
| at some points, which are of course called the boundary of the homology manifold. The boundary of an ''n''-dimensional [[first-countable]] homology manifold is an ''n''−1 dimensional homology manifold (without boundary).
| |
| | |
| ==Examples==
| |
| *Any topological manifold is a homology manifold.
| |
| *An example of a homology manifold that is not a manifold is the suspension of a [[homology sphere]] that is not a sphere.
| |
| *If ''X''×''Y'' is a topological manifolds, then ''X'' and ''Y'' are homology manifolds.
| |
| | |
| ==References==
| |
| *{{springer|id=H/h047800|title=Homology manifold|author=E. G. Sklyarenko}}
| |
| *W. J .R. Mitchell, "[http://links.jstor.org/sici?sici=0002-9939%28199010%29110%3A2%3C509%3ADTBOAH%3E2.0.CO%3B2-R Defining the boundary of a homology manifold]", [[Proceedings of the American Mathematical Society]], Vol. 110, No. 2. (Oct., 1990), pp. 509-513.
| |
| {{topology-stub}}
| |
| | |
| [[Category:Algebraic topology]]
| |
| [[Category:Generalized manifolds]]
| |
Hello. Let me introduce the author. Her name is Emilia Shroyer but it's not the most female title out there. To play baseball is the hobby he will never stop doing. Hiring is my profession. South Dakota is her beginning place but she needs to transfer simply because of her family.
My weblog - at home std testing