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

E.g., the theorem on preservation of domain is valid for , , the set is nowhere dense in if and only if , etc.

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

(cohomology with coefficients in a sheaf). Here is any integer; however, the homological dimension of the homology manifold over is , and thus the content of these isomorphisms is non-trivial only if . 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 of the sheaf and the group is immaterial. Instead of the group it is also possible to consider any locally constant sheaf of coefficients with stalk (this is accompanied by a change in ). Any open subset is a homology manifold. For this reason, the use of the equations

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

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

and

— 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

let be compact and let 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

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

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

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

References

[1]	E. Čech, "Théorie générale des variétés et de leurs théorèmes de dualité" Ann. of Math. (2) , 34 (1933) pp. 621–730
[2]	S. Lefschetz, "On generalized manifolds" Amer. J. Math. , 55 (1933) pp. 469–504
[3]	P.S. [P.S. Aleksandrov] Aleksandroff, "On local properties of closed sets" Ann. of Math. (2) , 36 : 1 (1935) pp. 1–35
[4]	P.S. [P.S. Aleksandrov] Aleksandroff, L.S. [L.S. Pontryagin] Pontrjagin, "Les variétés à dimensions généralisés" C.R. Acad. Sci. Paris Sér. I Math. , 202 (1936) pp. 1327–1329
[5]	P.A. Smith, "Transformations of finite period" Ann. of Math. (2) , 40 : 3 (1939) pp. 690–711
[6]	E.G. Begle, "Locally connected spaces and generalized manifolds" Amer. J. Math. , 64 (1942) pp. 553–574
[7]	R. Wilder, "Topology of manifolds" , Amer. Math. Soc. (1949)
[8]	A. Borel, "The Poincaré duality in generalized manifolds" Michigan Math. J. , 4 (1957) pp. 227–239
[9]	C.T. Yang, "Transformation groups on a homological manifold" Trans. Amer. Math. Soc. , 87 (1958) pp. 261–283
[10]	P.E. Conner, E.E. Floyd, "A characterization of generalized manifolds" Michigan Math. J. , 6 (1959) pp. 33–43
[11]	F. Raymond, "Separation and union theorems for generalized manifolds with boundary" Michigan Math. J. , 7 (1960) pp. 7–21
[12]	G.E. Bredon, "Orientation in generalized manifolds and application to the theory of transformation groups" Michigan Math. J. , 7 (1960) pp. 35–64
[13]	A. Borel, "Homology and duality in generalized manifolds" A. Borel (ed.) , Seminar on transformation groups , Princeton Univ. Press (1960) pp. 23–33
[14]	G.E. Bredon, "Wilder manifolds are locally orientable" Proc. Nat. Acad. Sci. USA , 63 : 4 (1969) pp. 1079–1081