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.