Homology group

of a topological space

A group which is associated to a topological space with the aim of conducting an algebraic study of the topological properties of the space. This correspondence should satisfy certain conditions, the most important of which are the Steenrod–Eilenberg axioms (see also Homology theory). Homology groups were originally based on ideas of H. Poincaré (1895) for polyhedra and their triangulations — representations as simplicial complexes (cf. Homology of a polyhedron). Subsequently the concept of homology was generalized and its domain of application was extended by producing a number of homology theories for arbitrary spaces in which the concept of a complex is used everywhere, but in a situation which is more involved than that of triangulation. There are two such fundamental theories: the singular and the spectral homology theory. The former is based on mappings of polyhedra into given spaces and is mostly applied to cases in which the polyhedra are mapped into arbitrary spaces, while the latter is based on mappings of arbitrary spaces into polyhedra, and is especially useful whenever the application in fact involves such mappings.

The idea of singular homology is due to O. Veblen (1921), who based his definition of homology of a space on systems consisting of polyhedra, their continuous mappings into the given space and their homology spaces. This idea resulted in the rise of two theories. Its direct development led to the group of continuous homology classes. The proper singular homology group, defined by S. Lefschetz (1933) and based on mappings of oriented simplexes into the given space, proved more useful, since it is defined on the base of groups of chains; subsequent development of this theory led to the study of ordered, rather than of oriented, simplexes by S. Eilenberg (1944) and to cubic homology theories in which cubes, rather than simplices, are employed (J.-P. Serre, 1951). All these kinds of singular homology groups are isomorphic under very general conditions.

Spectral homology, based on the homology of nerves of coverings of a space (cf. Nerve of a family of sets), connected with the spectrum by natural simplicial mappings of nerves, were introduced by P.S. Aleksandrov (1925–1928), who initially studied compact metric spaces and sequences of nerves of finite coverings. This theory was extended to arbitrary spaces with the aid of arbitrary systems of nerves of open coverings by E. Čech (1932), who also based himself on finite coverings, which is not always suitable in the case of non-compact spaces. For this reason infinite coverings began to be employed in the mid-forties. The homology group thus introduced is known as the Aleksandrov–Čech group (cf. Aleksandrov–Čech homology and cohomology). L. Vietoris (1927) gave another definition of the homology group for compact metric spaces, based on limit processes (cf. Vietoris homology). The definition of the Vietoris homology group for an arbitrary space is based on the study of complexes of coverings inscribed in each other (the so-called Vietoris complexes), the simplices of which are finite systems of points of the space which belong to the same element of the covering. The construction of cohomology groups based on cochains, which are functions of ordered sets of points of a space, was proposed in 1935 by A.N. Kolmogorov and J.W. Alexander, independently of each other. Kolmogorov also proposed a construction of a homology group based on set functions and dual to the preceding construction; this homology group is isomorphic, for any coefficient group, to the Steenrod homology group (cf. Steenrod duality) and, if the coefficient group is compact, to the Aleksandrov–Čech homology group. The Aleksandrov–Čech homology group and the Vietoris homology group are isomorphic. The Vietoris homology group and the Alexander–Kolmogorov cohomology group are, respectively, the inverse and the direct limit of dual spectra, given on the same spectrum of Vietoris complexes, and are thus dual. Depending on the homology groups taken on the nerves and on the Vietoris complexes in the construction of the respective spectral homology groups, two variants are obtained — projective and spectral. In the former case the groups selected are the homology groups of a chain complex which is the limit of the chain complexes of subcomplexes of finite nerves and, respectively, of Vietoris complexes; in the latter, the limits of the homology groups of these subcomplexes. In the case of a discrete coefficient group these groups are isomorphic; for cohomology groups the constructions are dual.

The singular and the spectral theories are isomorphic in the case of paracompact Hausdorff homologically locally connected spaces. The last property means that, for a given neighbourhood of each point it is possible to find a smaller neighbourhood for which the image under the imbedding homomorphism of the singular homology group in the homology group of the given neighbourhood is trivial (for integer homology groups of all dimensions; if the dimension is zero, reduced groups are meant); in other words, this means that each point is tautly imbedded in the space. This is a property of, for example, locally contractible spaces, in particular of polyhedra.

The properties by which these theories differ from one another are as follows. The singular (but not the spectral) theory has the property of having an exact homology sequence and is a homology with compact support. The spectral homology theory is exact if specified on the category of pairs of compact spaces and if the coefficient group is compact. It was originally developed for this very case. The spectral (but not the singular) theory has the continuity property, i.e. if a given compact pair is the inverse limit of the spectra of certain compact pairs, then the homology group of the given pair is the limit of the spectra of the homology groups of these pairs, and the tautness property, i.e. the homology group of a subspace is the limit of the spectra of the homology groups of its neighbourhoods. These theories also differ from each other by their excision properties. The singular theory is the unique homology theory with a given coefficient group on the category of CW-complexes with the additivity property: The homology group of the topological sum of spaces is the direct sum of the homology groups of the terms. The spectral theory is the only partially exact homology theory on the category of compact pairs with the continuity property.

Of the numerous other homology groups and cohomology groups and their generalizations one may also mention extraordinary homology theories, constructed by methods of homological algebra; homology and cohomology groups with coefficients in a sheaf; homology with local coefficients; homology groups of spectral type with an exact homology sequence; and homology groups modulo various proper subspaces.

References

[1]	N.E. Steenrod, S. Eilenberg, "Foundations of algebraic topology" , Princeton Univ. Press (1966)
[2]	P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960)
[3]	P.S. Aleksandrov, "Fundamental duality theorems for non-closed sets of an $n$-dimensional space" Mat. Sb. , 21 (63) : 2 (1947) pp. 161–232 (In Russian)
[4]	P.S. Aleksandrov, "On homological positioning properties of complexes and closed sets" Izv. Akad. Nauk SSSR Ser. Mat. , 6 (1942) pp. 227–282 (In Russian)
[5]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[6]	R. Godement, "Topologie algébrique et théorie des faisceaux" , Hermann (1958)
[7]	G.E. Bredon, "Sheaf theory" , McGraw-Hill (1967)
[8]	C. Teleman, "Grundzüge der Topologie und differenzierbare Mannigfaltigkeiten" , Deutsch. Verlag Wissenschaft. (1968) (Translated from Rumanian)
[9]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)