Vietoris-Begle theorem

One of the most important results in algebraic topology connecting homological (topological) characteristics of topological Hausdorff spaces , (cf. also Hausdorff space) and a continuous mapping ; it has applications, for example, in the fixed-point theory for mappings. There are variants of this theorem depending on the choice of the (co)homology functor (respectively, ) when studying homomorphisms (respectively, ; see [a1], [a2] and algebraic topology for the necessary constructions and definitions).

For the functor , where is a group of coefficients, one defines -acyclicity of a set by , for , for (and similarly for the functor ). If is -acyclic for all , then is said to be acyclic.

The simplest variant of the Vietoris–Begle theorem (close to [a3]) is as follows. Let , be compact Hausdorff spaces, let be the Aleksandrov–Čech homology functor (over the field of rational numbers; cf. also Aleksandrov–Čech homology and cohomology), let the mapping of compact pairs have non-empty acyclic pre-images for any and let ; then the induced homomorphisms , , are isomorphisms (cf. also Homomorphism; Isomorphism). This result is also valid if one drops the condition of compactness of spaces and pairs and replaces it by the condition that be a proper mapping (cf. also Proper morphism) [a4].

For the Aleksandrov–Kolmogorov functor in the category of paracompact Hausdorff spaces and a bounded continuous surjective mapping one studies the cohomology homomorphism , where is an -module. If the pre-image , for any , is -acyclic for all (for a fixed ), then the homomorphism is an isomorphism for and it is a monomorphism for [a2]. In the case of locally compact spaces , , the statement is valid for cohomologies with compact supports under the additional condition that be a proper mapping (cf. also Proper morphism) [a2].

In the case of metric spaces , , the requirement that the pre-images be -acyclic at all points can be weakened in that one allows sets for which the -acyclicity property is broken: (), (), where is the group of coefficients. One defines the relative dimension of in , as the supremum of , where runs over the subsets bounded in . One defines a "weight measure" of in by

If , then the homomorphism is [a5]:

for an epimorphism;

for an isomorphism; and

for a monomorphism. A mapping is said to be an -Vietoris mapping () if is a proper, surjective and for all [a4]. From the previous statement it follows that for an -Vietoris mapping the homomorphism is an isomorphism for . For a -Vietoris mapping, , for all , i.e. all the pre-images are acyclic; such mappings are called Vietoris mappings.

Fixed-point theory.

Vietoris–Begle-type theorems are connected with the problem of equality, , for some , with the problem of coincidence of pairs of mappings , and with the fixed-point problem for set-valued mappings (see, for example, [a6], [a7], [a8], [a4], [a9], [a10], [a11], [a13]).

In fact, the set-valued mapping , where is a surjection, gives a connection between the two problems: a point at which and coincide, defines a fixed point for (), and vice versa; in fact, if , then is equal to at any point .

For general set-valued mappings it is easy to construct a corresponding pair: consider the graph of the set-valued mapping ,

and its Cartesian projections , . One obtains the pair , for which a point of coincidence , , defines a fixed point of the set-valued mapping .

Topological characteristics.

Topological characteristics such as the Lefschetz number, the Kronecker characteristic, the rotation of the vector field (M.A. Krasnoselskii), the Brouwer–Hopf degree, are well known for single-valued mappings in finite-dimensional spaces (see, for example, [a12]). Analogous characteristics for general set-valued mappings have been constructed on the basis of homomorphisms , (respectively, ) of (co)homology groups of the pair for . These set-valued mappings satisfy the general conditions of compactness of images and have the property of upper semi-continuity. However, there is also a homological condition for a mapping to be -Vietoris, ensuring an isomorphism (respectively, ) in homology (cohomology) of dimension , and permitting one to construct a homomorphism (respectively, ), generated by the set-valued mapping in (co)homology by the formula (respectively, ).

S. Eilenberg and D. Montgomery [a6] have generalized the classical construction of the Lefschetz number to set-valued upper semi-continuous mappings with acyclic images, where is compact metric ANR-space:

here, is a canonical decomposition of , where are homomorphisms and is an isomorphism for any (due to the Vietoris–Begle theorem). If , then . This result was generalized by many authors (see [a9], [a10], [a4], [a7]). These generalizations involve weaker conditions of acyclicity, as well as certain different variants.

Degree theory.

To describe the topological characteristics of set-valued mappings like the degree or the Kronecker characteristics some definitions are needed. Let , , be separable topological spaces (cf. also Separable space), let be the space of compact subsets, and suppose the set-valued mapping is upper semi-continuous. Such a mapping is called

-acyclic if for all (here, is the set of points at which the -acyclicity of the images is broken);

-acyclic if it is -acyclic; this is equivalent to acyclicity of every image . A mapping is called generally -acyclic if there exist a space and single-valued continuous mappings , , where is -Vietoris and for all . The collection is then said to be a representation of the set-valued mapping , the pair is called a selecting pair, and the mapping is called a selector of . For an -acyclic mapping , the projections of the graph , give a selecting pair:

As an example, consider the main construction of the degree of a mapping from the unit disc in the Euclidean space under the condition that , where , is -acyclic, . A generalization of the Vietoris–Begle theorem given by E.G. Sklyarenko ensures the existence of cohomology isomorphisms , over the group . Then , where is given by the equality . Here, , respectively , is a generator of the group , respectively , which is isomorphic to , and (a construction given by D.G. Bourgin, L. Górniewicz, and others, see [a9], [a10], [a4]). If the mapping under consideration is generally acyclic, then for every selecting pair the set-valued mapping (the selector of ) is -acyclic, and for it , ; applying the previous construction for the selector of the set-valued mapping , one obtains for any selector . The generalized degree is the set generated by all selecting pairs for the -mapping . A more general construction (without the condition that be -acyclic) was introduced by B.D. Gelman (see [a10]); namely, the topological characteristic , where is defined by the equality

is a generator in , all the generators are in accordance with the orientation of , and is a connecting homomorphism.

Note that an earlier definition of rotation of a set-valued field , , with non-acyclic images was given in [a8], [a13], [a14].

References

[a1]	S. Eilenberg, N. Steenrod, "Foundations of algebraic topology" , Princeton Univ. Press (1952)
[a2]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966)
[a3]	E.G. Begle, "The Vietoris mappings theorem for bicompact spaces" Ann. of Math. , 51 : 2 (1950) pp. 534–550
[a4]	L. Górniewicz, "Homological methods in fixed-point theory of multi-valued maps" Dissert. Math. , CXXIX (1976) pp. 1–71
[a5]	E.G. Sklyarenko, "Of some applications of theory of bundles in general topology" Uspekhi Mat. Nauk , 19 : 6 (1964) pp. 47–70 (In Russian)
[a6]	S. Eilenberg, D. Montgomery, "Fixed point theorems for multi-valued transformations" Amer. J. Math. , 68 (1946) pp. 214–222
[a7]	A. Granas, J.W. Jaworowski, "Some theorems on multi-valued maps of subsets of the Euclidean space" Bull. Acad. Polon. Sci. , 7 : 5 (1959) pp. 277–283
[a8]	Yu.G. Borisovich, B.D. Gelman, V.V. Obukhovskii, "Of some topological invariants of set-valued maps with nonconvex images" Proc. Sem. Functional Analysis, Voronezh State Univ. , 12 (1969) pp. 85–95
[a9]	Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis, V.V. Obukhovskii, "Topological methods in the fixed-point theory of multi-valued maps" Russian Math. Surveys , 35 : 1 (1980) pp. 65–143 (In Russian)
[a10]	Yu.G. Borisovich, B.D. Gelman, A.D. Myshkis, V.V. Obukhovskii, "Multivalued mappings" J. Soviet Math. , 24 (1984) pp. 719–791 (In Russian)
[a11]	Yu.G. Borisovich, "A modern appoach to the theory of topological characteristics of nonlinear operators II" , Global analysis: Studies and Applications IV , Lecture Notes Math. , 1453 , Springer (1990) pp. 21–49
[a12]	Yu.G. Borisovich, N.M. Bliznyakov, T.N. Fomenko, Y.A. Izrailevich, "Introduction to differential and algebraic topology" , Kluwer Acad. Publ. (1995)
[a13]	L. Górniewicz, "On non-acyclic multi-valued mappings of subsets of Euclidean spaces" Bull. Acad. Polon. Sci. , 20 : 5 (1972) pp. 379–385
[a14]	D.G. Bouvgin, "Cones and Vietoris–Begle type theorems" Trans. Amer. Math. Soc. , 174 (1972) pp. 155–183