Homology theory

of topological spaces

A part of algebraic topology which realizes a connection between topological and algebraic concepts. By associating to each space a certain sequence of groups, and to each continuous mapping of spaces, homomorphisms of the respective groups, homology theory uses the properties of groups and their homomorphisms to clarify the properties of spaces and mappings. Such properties include, for example, the connections between various dimensionalities, the study of which is based on the concept of excision, unlike the other part of algebraic topology — the theory of homotopy, in which deformations are used for the same purpose. Homology theory was introduced towards the end of the 19th century by H. Poincaré (cf. Homology of a polyhedron), but the axiomatic construction (including the precise limits of this concept, which had been indefinite for a long time) was imparted to it only by S. Eilenberg and N. Steenrod [3] (cf. Algebraic topology; Homology group; Steenrod–Eilenberg axioms).

According to this construction a homology theory consists of three functions: 1) relative -dimensional homology groups of a pair of topological spaces , , which assign to each pair and each integer an Abelian group ; 2) the homomorphism

which is assigned to a continuous mapping and a number , and which is called the homomorphism induced by ; and 3) the boundary operator , which assigns to each pair and each a homomorphism of into (the so-called absolute group of the space , which is the group of the pair ). These functions must satisfy the following axioms.

1) If is the identity mapping, so is .

2) .

3) .

4) The exactness axiom. If

are the natural inclusions, then the sequence

the so-called homology sequence of the pair , is exact, that is, the image of each incoming homomorphism equals the kernel of the outgoing one.

5) The homotopy axiom. If the mappings

are homotopic, then .

6) The excision axiom. If is an open subset of the space and its closure is contained in the interior of the subspace , then the inclusion

induces an isomorphism .

7) The dimension axiom. If is a one-point space, then for all .

Instead of taking the category of all pairs of spaces as the domain of definition of , it is also possible to take an arbitrary category of pairs of spaces, e.g. the category of pairs of compact spaces or the category of pairs consisting of polyhedra and their subpolyhedra. However, such a category must contain along with also the pairs , , , , , the cylinder , where , and some one-point space , with all their inclusions. Such a category must also include all pairs and mappings which are encountered in the axioms or in the theorems. On the other hand, instead of taking the category of all Abelian groups as the range of , it is also possible to use other categories, e.g. the category of topological, in particular that of compact, groups with continuous homomorphisms, or the category of modules over some ring with linear homomorphisms.

Axioms 1 and 2 mean that is a covariant functor from some category of pairs of spaces into the category of groups. Axiom 3 means that the boundary operator is a natural transformation of the functor to the functor . Axiom 4, which connects the functors of all dimensions , is sometimes replaced by the weaker requirement: The sequence should only be partially exact, i.e. the image should be included in the kernel (cf. Exact sequence); an important example of a partially exact homology theory is the Aleksandrov–Čech homology theory. Axiom 5 may be written in an equivalent form: If

are mappings defined by the formulas , , then . Axiom 6, which requires the invariance under excision and which has a number of different formulations, displays the property of homology theory by which it differs from homotopy theory. Axiom 7, which ensures the geometric meaning of the dimensionality index , is often omitted in modern studies. One then obtains so-called generalized homology theories, an important example of which is bordism theory.

There exists a cohomology theory dual to a homology theory (cf. Duality in topology). It is given by relative -dimensional cohomology groups , which are contravariant functors from the category of pairs of topological spaces into the category of Abelian groups, with induced homomorphisms

and coboundary operators

The axioms are formulated in the same manner as for homology, with the obvious reversal of the direction of the homomorphisms. For instance, the exactness axiom requires the existence of an exact cohomology sequence

And, analogously, there are also generalized cohomology theories, important examples of which are -theory and cobordism theory. The facts given below concerning homology theory have cohomology analogues.

The coefficient group of a homology or a cohomology theory is the group or, respectively, . The so-called reduced groups are often conveniently substituted for the groups : the reduced zero-dimensional homology group is the kernel of the homomorphism

induced by the mapping , while the reduced zero-dimensional cohomology group is the quotient group of by the image ; the reduced groups of other dimensions are identical with the initial ones: , . Thus, . If , then for all . The replacement of the ordinary by the reduced groups permits one to convert the homology sequence into a reduced homology sequence.

The axioms of homology theory are not independent. E.g., axiom 1 is a consequence of the axioms 2, 3 and 4. The system of axioms is compatible, as is seen from the example of the trivial theory ; non-trivial examples include the Aleksandrov–Čech cohomology theory, singular homology, etc. Regarding uniqueness, the following holds: A homomorphism of a homology theory into a homology theory is a system of homomorphisms

such that

and

If all are isomorphisms, then the homology theories and are called isomorphic. A homology theory on finite polyhedra is uniquely determined by its coefficient group. More precisely, if is an arbitrary homomorphism of the coefficient group of into the coefficient group of , then for each polyhedral pair there exists a unique homomorphism

with the property . If is an isomorphism, so are all . Since the homology groups of negative dimension of a triangulable pair are trivial, the equality , , is valid for any homology theory as well. The uniqueness theorem is also valid for wider categories of spaces if the homology theory satisfies appropriate additional axioms.

The homology groups are topological and also homotopy invariants: If is a homotopy equivalence, then is an isomorphism. If is a contractible space — a cell, in particular — then , , and . If is a homotopy equivalence, then and, for any , . If is a retract of the space , then is a monomorphism, is an epimorphism, the operation is trivial, and

If is deformable into , then is an epimorphism, is trivial, is a monomorphism, and

Let denote the suspension over . The following isomorphism is then valid:

This makes it possible to compute the homology groups of the spheres : if , and ; consequently, if ; if or ; and .

An important role in homology theory is played by homology sequences of triples and triads. In the case of a triple , , of spaces, the boundary operator is defined as the composition , where is the inclusion. There results the so-called homology sequence of the triple (which, for , reduces to the homology sequence of the pair ):

where and are the inclusions. This sequence is exact. If the groups , respectively , , are trivial for all , then , respectively , , are isomorphisms, and vice versa. If is the union of non-intersecting closed sets , , and , where , then is isomorphic to the direct sum of the groups , . A triad is a space with an ordered pair of subspaces (cf. also Triads). It is a proper triad if the inclusions

induce isomorphisms, or if the decomposition

is valid. Further, the boundary operator

is defined as , where . This generates the exact homology sequence of the triad:

where , are the inclusions (if , this sequence reduces to the homology sequence of the triple ).

Let , and suppose the mappings satisfy the relations , , , . The following addition theorems are then valid.

1) .

2) If is contractible and if are defined, respectively, by means of and , then the equality holds for the induced homomorphisms of the reduced groups .

Define the homomorphism

by , , where , are the inclusions. Similarly, define

by , , where , are the inclusions. Finally, define

by , where

are the inclusions. One then obtains the so-called Mayer–Vietoris sequence of the proper triad:

which is exact and which relates the homology groups of the spaces and with the homology groups of their union and intersection. Then, if , one may pass to a similar sequence for the reduced groups, which implies that:

1) if is contractible, then

2) if is contractible, then

3) if both and are contractible, then defines an isomorphism

These results make it possible to compute the homology groups of various spaces. For instance, if is a closed orientable surface of genus , then is isomorphic to the coefficient group if ; to the direct sum of copies of if ; and to 0 in the remaining cases. If is a closed non-orientable surface of genus , then is isomorphic to if ; to , where is the quotient group , , if ; to the subgroup of consisting of all elements with if ; and to 0 in the remaining cases. Thus, homology theory gives a topological classification of closed surfaces. For an -dimensional real projective space the group is isomorphic to if or and odd; to if is odd and ; to if is even and ; and to 0 in the remaining cases. The homology group of the complex projective space of dimension is isomorphic to if is even and ; and to 0 in the remaining cases. The homology group of the lens space is isomorphic to if ; to , where , if ; to , where , if ; and to 0 in the remaining cases.

From the various applications of the results described above certain fundamental theorems are stated here. First of all, the invariance of dimension: spheres, as well as Euclidean spaces, of different dimensions are not homeomorphic; in fact, if two polyhedra are homeomorphic, then they have the same dimension. Furthermore, the equality , where is an extension of a given mapping , , yields various criteria of extendability and retractibility of mappings; for example, a mapping of a sphere , , of non-zero degree into itself is not extendable to the -dimensional ball with boundary , and is not a retract of for any natural . This, in turn, yields Brouwer's fixed-point theorem: Any continuous mapping has a fixed point. Finally, it may be proved that a unit tangent vector field exists on if and only if is odd, while the theory of triads yields several theorems on the degrees of mappings; in particular, it is possibly to give a new proof of the fundamental theorem of algebra.

References

[1]	P.S. Aleksandrov, "Combinatorial topology" , Graylock , Rochester (1956) (Translated from Russian)
[2]	S. Lefschetz, "Algebraic topology" , Amer. Math. Soc. (1942)
[3]	S. Eilenberg, N.E. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952)
[4]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[5]	S.-T. Hu, "Homology theory" , Holden-Day (1966)
[6]	S.-T. Hu, "Cohomology theory" , Markham , Chicago (1968)
[7]	A. Dold, "Lectures on algebraic topology" , Springer (1980)

Comments

A sequence of modules , , over a ring together with homomorphisms such that for all is often called a chain complex. Thus, in case of a partially exact homology theory the sequence of a pair is a chain complex, or complex, instead of a long exact sequence. Dually one has cochain complexes . Many homology and cohomology theories are constructed via chain and cochain complexes: first to a pair there is associated a chain complex and then , and analogously for cohomology.

References

[a1]	R.M. Switzer, "Algebraic topology - homotopy and homology" , Springer (1975)
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. 113ff
[a3]	S. Lefschetz, "Topology" , Chelsea, reprint (1956)
[a4]	C.R.F. Maunder, "Algebraic topology" , v. Nostrand-Reinhold (1970)
[a5]	I. Vaisman, "Cohomology and differential forms" , M. Dekker (1973)