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 $ n $- manifold (a generalized $ n $- manifold) over a group or a module $ G $ of coefficients is a locally compact topological space $ X $ with finite homological dimension (cf. Homological dimension of a space) over $ G $ and such that all its local homology groups (cf. Local homology) $ H _ {p} ^ {x} $ are trivial if $ p \neq n $, and are isomorphic either to $ G $ or zero if $ p = n $. Here, $ H _ {p} ^ {x} $ is the direct limit of the groups $ H _ {p} ( X, X \setminus U; G ) $, taken over all neighbourhoods $ U $ of the point $ x \in X $, and $ H $ is a homology theory that satisfies all the Steenrod–Eilenberg axioms, including the exactness axiom. In the category of locally contractible spaces the theory $ H $, considered with compact support, is isomorphic to the singular theory (cf. Singular homology). The groups $ H _ {n} ^ {x} $ automatically turn out to be the stalks of some sheaf $ {\mathcal H} _ {n} $( cf. Sheaf theory), known as the orienting sheaf of the manifold $ X $. A homology manifold $ X $ is said to be orientable if the sheaf $ {\mathcal H} _ {n} $ is isomorphic to the constant sheaf $ X \times G $, and is said to be locally orientable if $ {\mathcal H} _ {n} $ is locally constant at the points where $ H _ {n} ^ {x} \neq 0 $. If $ G $ is a principal ideal ring and if all $ H _ {n} ^ {x} $ are non-zero, a homology manifold over $ G $ is always locally orientable. If a homology manifold over a group $ G $ is locally orientable, then the set of all $ x \in X $ on which $ H _ {n} ^ {x} = 0 $ is closed, nowhere dense and forms the boundary of the homology manifold $ X $. A locally orientable homology manifold $ X $ has the same homological properties as ordinary manifolds.

E.g., the theorem on preservation of domain is valid for $ X $, $ h \mathop{\rm dim} _ {G} X = n $, the set $ A ^ \prime $ is nowhere dense in $ X $ if and only if $ h \mathop{\rm dim} _ {G} A \leq n - 1 $, etc.

For any homology manifold over $ G $ there are natural isomorphisms (Poincaré duality)

$$ H _ {p} ( X; G) = H ^ {n - p } ( X; {\mathcal H} _ {n} ) $$

(cohomology with coefficients in a sheaf). Here $ p $ is any integer; however, the homological dimension of the homology manifold $ X $ over $ G $ is $ n $, and thus the content of these isomorphisms is non-trivial only if $ 0 \leq p \leq n $. 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 $ H _ {n} ^ {x} $ of the sheaf $ {\mathcal H} _ {n} $ and the group $ G $ is immaterial. Instead of the group $ G $ it is also possible to consider any locally constant sheaf of coefficients $ {\mathcal G} $ with stalk $ G $( this is accompanied by a change in $ {\mathcal H} _ {n} $). Any open subset $ U \subset X $ is a homology manifold. For this reason, the use of the equations

$$ H _ {p} ( U; G) = H _ {p} ( X, X \setminus U; G),\ p \neq 0, n, $$

$$ H _ {c} ^ {q} ( U; G) = H ^ {q} ( X, X \setminus U; G), $$

in the second one of which $ U $ has compact closure, while the index $ c $ indicates the compactness of the supports, makes it possible to obtain the isomorphisms

$$ H _ {p} ( X, X \setminus U; G) = H ^ {n - p } ( U, {\mathcal H} _ {n} ), $$

$$ H _ {p} ^ {c} ( U; G) = H ^ {n - p } ( X, X \setminus U; {\mathcal H} _ {n} ) $$

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

$$ H _ {p} ( X \setminus U; G) = H ^ {n - p } ( X, U; {\mathcal H} _ {n} ) $$

and

$$ H _ {p} ( X, U; G) = H ^ {n - p } ( X \setminus U; {\mathcal H} _ {n} ) $$

— the latter one being a generalization of 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

$$ H _ {p} ( X; G) = H _ {p + 1 } ( X; G) = 0 , $$

let $ X $ be compact and let $ Y $ 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

$$ H _ {p} ^ {c} ( X \setminus Y; G) = H ^ {n - p - 1 } ( Y; {\mathcal H} _ {n} ), $$

which represents the Pontryagin duality for a closed $ Y $ and the Steenrod duality for an open $ Y $. This and the property of continuity of cohomology implies that the isomorphism

$$ H _ {p} ^ {c} ( X \setminus Y; G) = H ^ {n - p- 1 } ( Y; {\mathcal H} _ {n} ) $$

is valid for any subset $ Y \subset X $( Sitnikov duality). If $ X $ is non-compact one must consider homology with supports closed in all of $ X $ rather than homology with compact supports. If $ X $ is compact, reduced homology must be used for $ p = 0 $.

Non-trivial examples of homology manifolds include "factors" of ordinary manifolds such as Euclidean spaces: If for a topological space $ X $ there exists an $ Y $ such that the Cartesian product $ X \times Y $ is a homology manifold, then $ X $ and $ Y $ 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), where they appear as 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 $ G $, and if $ G $ is at most countable, then the converse proposition is true as well.

References

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