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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 |
Vietoris-Begle theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vietoris-Begle_theorem&oldid=15219