Vietoris-Begle theorem

One of the most important results in algebraic topology connecting homological (topological) characteristics of topological Hausdorff spaces $X$, $Y$ (cf. also Hausdorff space) and a continuous mapping $f : X \rightarrow Y$; 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 $H_{*}$ (respectively, $H ^ { * }$) when studying homomorphisms $f _{*} : H * ( X ) \rightarrow H_{ *} ( Y )$ (respectively, $f ^ { * } : H ^ { * } ( Y ) \rightarrow H ^ { * } ( X )$; see [a1], [a2] and algebraic topology for the necessary constructions and definitions).

For the functor $H _ * (\, . \, ; G )$, where $G$ is a group of coefficients, one defines $q$-acyclicity of a set $M \subset X$ by $H _ { q } ( M , G ) = 0$, for $q > 0$, $H _ { 0 } ( M , G ) \cong G$ for $q = 0$ (and similarly for the functor $H ^ { * }$). If $M$ is $q$-acyclic for all $q \geq 0$, then $M$ is said to be acyclic.

The simplest variant of the Vietoris–Begle theorem (close to [a3]) is as follows. Let $X$, $Y$ be compact Hausdorff spaces, let $H_{*} ( X , \mathbf{Q} )$ be the Aleksandrov–Čech homology functor (over the field $\mathbf{Q}$ of rational numbers; cf. also Aleksandrov–Čech homology and cohomology), let the mapping of compact pairs $f : ( X , X _ { 0 } ) \rightarrow ( Y , Y _ { 0 } )$ have non-empty acyclic pre-images $f ^ { - 1 } ( y )$ for any $y \in Y$ and let $f ^ { - 1 } ( Y _ { 0 } ) = X _ { 0 }$; then the induced homomorphisms $f_{*} : H _ { q } ( X , X _ { 0 } ) \rightarrow H _ { q } ( Y , Y _ { 0 } )$, $q \geq 0$, 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 $f$ be a proper mapping (cf. also Proper morphism) [a4].

For the Aleksandrov–Kolmogorov functor $\overline { H } \square ^ { * }$ in the category of paracompact Hausdorff spaces and a bounded continuous surjective mapping $f : X \rightarrow Y$ one studies the cohomology homomorphism $f ^ { * } : \overline { H } \square ^ { * } ( Y , G ) \rightarrow \overline { H } \square ^ { * } ( X , G )$, where $G$ is an $\mathbf{R}$-module. If the pre-image $f ^ { - 1 } ( y )$, for any $y \in Y$, is $q$-acyclic for all $q < n$ (for a fixed $n > 0$), then the homomorphism $f ^ { * } : \overline { H } \square ^ { q } ( Y , G ) \rightarrow \overline { H } \square ^ { q } ( X , G )$ is an isomorphism for $q < n$ and it is a monomorphism for $q = n$ [a2]. In the case of locally compact spaces $X$, $Y$, the statement is valid for cohomologies $\overline { H } \square _ { c } ^ { * }$ with compact supports under the additional condition that $f$ be a proper mapping (cf. also Proper morphism) [a2].

In the case of metric spaces $X$, $Y$, the requirement that the pre-images $f ^ { - 1 } ( y )$ be $q$-acyclic at all points $y \in Y$ can be weakened in that one allows sets $M _ { k } ( f ) \subset Y$ for which the $k$-acyclicity property is broken: $H ^ { k } ( f ^ { - 1 } ( y ) , G ) \neq 0$ ($k > 0$), $H ^ { 0 } ( f ^ { - 1 } ( y ) , G ) \notin G$ ($k = 0$), where $G$ is the group of coefficients. One defines the relative dimension of $M _ { k }$ in $Y$, $d _ { k } = \operatorname{rd} _ { Y } M _ { k }$ as the supremum of $\operatorname { dim } Q$, where $Q \subset M _ { k }$ runs over the subsets bounded in $Y$. One defines a "weight measure" of $M _ { k }$ in $Y$ by

\begin{equation*} \nu = \operatorname { max } _ { 0 \leq k \leq N - 1 } ( d _ { k } + k ). \end{equation*}

If $\nu < N - 1$, then the homomorphism $f ^ { * } : H ^ { q } ( Y , G ) \rightarrow H ^ { q } ( X , G )$ is [a5]:

for $q = \nu + 1$ an epimorphism;

for $\nu + 1 < q < N$ an isomorphism; and

for $q = N$ a monomorphism. A mapping $f : X \rightarrow Y$ is said to be an $n$-Vietoris mapping ($n \geq 1$) if $f$ is a proper, surjective and $\operatorname { rd } _{Y} ( M _ { k } (\, f ) ) \leq n - 2 - k $ for all [a4]. From the previous statement it follows that for an $n$-Vietoris mapping $f : X \rightarrow Y$ the homomorphism $f ^ { * } : H ^ { q } ( Y , G ) \rightarrow H ^ { q } ( X , G )$ is an isomorphism for $q \geq n$. For a $1$-Vietoris mapping, $H ^ { 0 } ( f ^ { - 1 } ( y ) , G ) = G , H ^ { q } ( f ^ { - 1 } ( y ) , G ) = 0$, for all $q > 0$, i.e. all the pre-images $f ^ { - 1 } ( y )$ are acyclic; such mappings are called Vietoris mappings.

Fixed-point theory.

Vietoris–Begle-type theorems are connected with the problem of equality, $f ( x ) = g ( x )$, for some $x$, with the problem of coincidence of pairs $( f , g )$ of mappings $f , g : X \rightarrow Y$, 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 $G = f \circ g ^ { - 1 } : Y \rightarrow Y$, where $g$ is a surjection, gives a connection between the two problems: a point $x _ { 0 }$ at which $f$ and $g$ coincide, defines a fixed point $y _ { 0 } = g ( x _ { 0 } )$ for $G$ ($y _ { 0 } \in G ( y _ { 0 } )$), and vice versa; in fact, if $y _ { 0 } \in \operatorname{Fix} G$, then $f$ is equal to $g$ at any point $x _ { 0 } \in g ^ { - 1 } ( y _ { 0 } )$.

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

\begin{equation*} \Gamma ( F ) = \{ ( x , y ) \in X \times X : y \in F ( x ) \} \end{equation*}

and its Cartesian projections $p ( x , y ) = x$, $q ( x , y ) = y$. One obtains the pair $( p , q ) : \Gamma ( F ) \rightarrow X$, for which a point of coincidence $( x _ { 0 } , y _ { 0 } ) \in \Gamma ( F )$, $p ( x _ { 0 } , y _ { 0 } ) = q ( x _ { 0 } , y _ { 0 } )$, defines a fixed point $x _ { 0 } \in F ( x _ { 0 } )$ of the set-valued mapping $F$.

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 $F$ have been constructed on the basis of homomorphisms $( p _ {*} , q _ { * } )$, (respectively, $( p ^ { * } , q ^ { * } )$) of (co)homology groups of the pair $( p , q )$ for $F$. 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 $F$ to be $n$-Vietoris, ensuring an isomorphism $p_{ *}$ (respectively, $p ^ { * }$) in homology (cohomology) of dimension $q \geq n$, and permitting one to construct a homomorphism $F_{*}$ (respectively, $F ^ { * }$), generated by the set-valued mapping in (co)homology by the formula $F * = q * p * ^ { - 1 }$ (respectively, $F ^ { * } = p ^ { * - 1} q ^ { * }$).

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

\begin{equation*} \Lambda ( F ) = \sum _ { n = 0 } ^ { \infty } ( - 1 ) ^ { n } \operatorname { tr } ( r _n* \circ t_n * ^ { - 1 } ); \end{equation*}

here, $F = r \circ t ^ { - 1 }$ is a canonical decomposition of $F$, where are homomorphisms and $t _ { n_{*} }$ is an isomorphism for any $n \geq 0$ (due to the Vietoris–Begle theorem). If $\Lambda ( F ) \neq \theta$, then $\operatorname {Fix} F \neq \emptyset$. 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 $F$ like the degree $\operatorname { deg } F$ or the Kronecker characteristics $\gamma ( F )$ some definitions are needed. Let $X$, $Y$, $Z$ be separable topological spaces (cf. also Separable space), let $K ( Y )$ be the space of compact subsets, and suppose the set-valued mapping $F : X \rightarrow K ( Y )$ is upper semi-continuous. Such a mapping is called

$n$-acyclic if $\operatorname { rd }_{X} ( N _ { K } ( F ) ) \leq n - k - 2$ for all (here, $N _ { K } ( F ) \subset X$ is the set of points $x$ at which the $k$-acyclicity of the images $F ( x )$ is broken);

$F$-acyclic if it is $1$-acyclic; this is equivalent to acyclicity of every image $F ( x )$. A mapping $F : X \rightarrow K ( Y )$ is called generally $n$-acyclic if there exist a space $Z$ and single-valued continuous mappings $p : Z \rightarrow X$, $q : Z \rightarrow Y$, where $p$ is $n$-Vietoris and $q \circ p ^ { - 1 } ( x ) \subset F ( x )$ for all $x \in X$. The collection $\{ X , Y , Z , p , q \}$ is then said to be a representation of the set-valued mapping $F$, the pair $( p , q )$ is called a selecting pair, and the mapping $q \circ p ^ { - 1 }$ is called a selector of $F$. For an $n$-acyclic mapping $F$, the projections of the graph $t : X \times Y \supset \Gamma ( F ) \rightarrow X$, $r : X \times Y \supset \Gamma ( F ) \rightarrow Y$ give a selecting pair:

\begin{equation*} F ( x ) = r \circ t ^ { - 1 } ( x ). \end{equation*}

As an example, consider the main construction of the degree of a mapping $F : \overline { D } \square ^ { n + 1 } \rightarrow K ( E ^ { n + 1 } )$ from the unit disc $\overline { D } \square ^ { n + 1 } \subset E ^ { n + 1 }$ in the Euclidean space $E ^ { n + 1}$ under the condition that $F : S ^ { n } \rightarrow K ( E ^ { n + 1 } \backslash \theta )$, where $S ^ { n } = \partial \overline { D } \square ^ { n + 1 }$, is $m$-acyclic, $1 \leq m \leq n$. A generalization of the Vietoris–Begle theorem given by E.G. Sklyarenko ensures the existence of cohomology isomorphisms $t ^ { * } : H ^ { n } ( S ^ { n } ) \rightarrow H ^ { n } ( \Gamma _ { S ^ { n } } )$, $\hat { t } \square ^ { * } : H ^ { n + 1 } ( \overline { D } \square ^ { n + 1 } , S ^ { n } ) \rightarrow H ^ { n + 1 } ( \Gamma _ { \overline{D} \square ^ { n + 1 } } , \Gamma _ { S ^ { n } } )$ over the group $\bf Z$. Then $\operatorname { deg } ( F , \overline { D } \square ^ { n + 1 } , \theta ) = k$, where $k$ is given by the equality $( t ^ { * } ) ^ { - 1 } \circ ( t - r ) ^ { * } \beta _ { 1 } = k \beta _ { 2 }$. Here, $\beta _ { 1 }$, respectively $\beta_2$, is a generator of the group $H ^ { n } ( E ^ { n + 1 } \backslash \theta )$, respectively $H ^ { n } ( S ^ { n } )$, which is isomorphic to $\bf Z$, and $( t - r ) : ( \Gamma _ { S ^ { n } } ) \rightarrow ( E ^ { n + 1 } \backslash 0 )$ (a construction given by D.G. Bourgin, L. Górniewicz, and others, see [a9], [a10], [a4]). If the mapping $F : X \rightarrow K ( Y )$ under consideration is generally acyclic, then for every selecting pair $( p , q ) \subset F$ the set-valued mapping (the selector of $F$) $G = p \circ q ^ { - 1 } : X \rightarrow K ( Y )$ is $m$-acyclic, and for it $r = p$, $t = q$; applying the previous construction for the selector $G$ of the set-valued mapping $F : ( \overline { D } \square ^ { n + 1 } , S ^ { n } ) \rightarrow ( K ( E ^ { n + 1 } ) , K ( E ^ { n + 1 } \backslash \theta ) )$, one obtains $\operatorname { deg } ( G , \overline { D } \square ^ { n + 1 } , \theta )$ for any selector $G$. The generalized degree $\operatorname { Deg } ( F , \overline { D } \square ^ { n + 1 } , \theta )$ is the set $\{ \operatorname { deg } ( G , \overline { D } \square ^ { n + 1 } , \theta ) \}$ generated by all selecting pairs $( p , q )$ for the $n$-mapping $F$. A more general construction (without the condition that $F$ be $m$-acyclic) was introduced by B.D. Gelman (see [a10]); namely, the topological characteristic $\kappa ( F , \overline { D } \square ^ { n + 1 } ) = k$, where $k$ is defined by the equality

\begin{equation*} \delta ^ { * } \circ ( t - r ) ^ { * } \beta _ { 1 } = k ( \widehat{t ^ { * }} \square ^ { - 1 } \beta _ { 3 } ), \end{equation*}

$\beta_3$ is a generator in $H ^ { n + 1 } ( \overline { D } \square ^ { n + 1 } , S ^ { n } ) \cong \mathbf{Z}$, all the generators $\beta _ { i }$ are in accordance with the orientation of $E ^ { n + 1}$, and $\delta ^ { * } : H ^ { n } ( \Gamma _ { S ^ { n } } ) \rightarrow H ^ { n + 1 } ( \Gamma _ { \bar{D} \square ^ { n + 1 } } , \Gamma _ { S ^ { n } } )$ is a connecting homomorphism.

Note that an earlier definition of rotation of a set-valued field $\Phi x = x - F x$, $\Phi : \partial U \rightarrow E ^ { n + 1 } {\color{blue} \backslash} 0$, 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