View source for Vietoris-Begle theorem

One of the most important results in [[Algebraic topology|algebraic topology]] connecting homological (topological) characteristics of topological Hausdorff spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200201.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200202.png" /> (cf. also [[Hausdorff space|Hausdorff space]]) and a [[Continuous mapping|continuous mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200203.png" />; 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200204.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200205.png" />) when studying homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200206.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200207.png" />; see [[#References|[a1]]], [[#References|[a2]]] and [[Algebraic topology|algebraic topology]] for the necessary constructions and definitions).

For the [[Functor|functor]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200208.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v1200209.png" /> is a group of coefficients, one defines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002011.png" />-acyclicity of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002012.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002013.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002015.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002016.png" /> (and similarly for the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002017.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002018.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002019.png" />-acyclic for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002020.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002021.png" /> is said to be acyclic.

The simplest variant of the Vietoris–Begle theorem (close to [[#References|[a3]]]) is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002023.png" /> be compact Hausdorff spaces, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002024.png" /> be the Aleksandrov–Čech homology functor (over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002025.png" /> of rational numbers; cf. also [[Aleksandrov–Čech homology and cohomology|Aleksandrov–Čech homology and cohomology]]), let the mapping of compact pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002026.png" /> have non-empty acyclic pre-images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002027.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002028.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002029.png" />; then the induced homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002031.png" />, are isomorphisms (cf. also [[Homomorphism|Homomorphism]]; [[Isomorphism|Isomorphism]]). This result is also valid if one drops the condition of compactness of spaces and pairs and replaces it by the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002032.png" /> be a proper mapping (cf. also [[Proper morphism|Proper morphism]]) [[#References|[a4]]].

For the Aleksandrov–Kolmogorov functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002033.png" /> in the category of paracompact Hausdorff spaces and a bounded continuous surjective mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002034.png" /> one studies the cohomology homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002036.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002037.png" />-module. If the pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002038.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002039.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002040.png" />-acyclic for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002041.png" /> (for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002042.png" />), then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002043.png" /> is an isomorphism for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002044.png" /> and it is a [[Monomorphism|monomorphism]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002045.png" /> [[#References|[a2]]]. In the case of locally compact spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002047.png" />, the statement is valid for cohomologies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002048.png" /> with compact supports under the additional condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002049.png" /> be a proper mapping (cf. also [[Proper morphism|Proper morphism]]) [[#References|[a2]]].

In the case of metric spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002051.png" />, the requirement that the pre-images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002052.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002053.png" />-acyclic at all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002054.png" /> can be weakened in that one allows sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002055.png" /> for which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002056.png" />-acyclicity property is broken: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002057.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002058.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002059.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002060.png" />), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002061.png" /> is the group of coefficients. One defines the relative dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002064.png" /> as the supremum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002065.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002066.png" /> runs over the subsets bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002067.png" />. One defines a  "weight measure"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002068.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002069.png" /> by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002070.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002071.png" />, then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002072.png" /> is [[#References|[a5]]]:

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002073.png" /> an [[Epimorphism|epimorphism]];

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002074.png" /> an [[Isomorphism|isomorphism]]; and

for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002075.png" /> a [[Monomorphism|monomorphism]]. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002076.png" /> is said to be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002078.png" />-Vietoris mapping (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002079.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002080.png" /> is a proper, surjective and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002081.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002082.png" /> [[#References|[a4]]]. From the previous statement it follows that for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002083.png" />-Vietoris mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002084.png" /> the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002085.png" /> is an isomorphism for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002086.png" />. For a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002087.png" />-Vietoris mapping, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002088.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002089.png" />, i.e. all the pre-images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002090.png" /> are acyclic; such mappings are called Vietoris mappings.

==Fixed-point theory.==
Vietoris–Begle-type theorems are connected with the problem of equality, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002091.png" />, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002092.png" />, with the problem of coincidence of pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002093.png" /> of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002094.png" />, and with the fixed-point problem for set-valued mappings (see, for example, [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a4]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]], [[#References|[a13]]]).

In fact, the set-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002096.png" /> is a surjection, gives a connection between the two problems: a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002097.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v12002099.png" /> coincide, defines a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020100.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020101.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020102.png" />), and vice versa; in fact, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020103.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020104.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020105.png" /> at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020106.png" />.

For general set-valued mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020107.png" /> it is easy to construct a corresponding pair: consider the graph of the set-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020108.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020109.png" /></td> </tr></table>

and its Cartesian projections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020110.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020111.png" />. One obtains the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020112.png" />, for which a point of coincidence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020114.png" />, defines a fixed point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020115.png" /> of the set-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020116.png" />.

==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, [[#References|[a12]]]). Analogous characteristics for general set-valued mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020117.png" /> have been constructed on the basis of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020118.png" />, (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020119.png" />) of (co)homology groups of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020120.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020121.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020122.png" /> to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020123.png" />-Vietoris, ensuring an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020124.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020125.png" />) in homology (cohomology) of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020126.png" />, and permitting one to construct a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020127.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020128.png" />), generated by the set-valued mapping in (co)homology by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020129.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020130.png" />).

S. Eilenberg and D. Montgomery [[#References|[a6]]] have generalized the classical construction of the Lefschetz number to set-valued upper semi-continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020131.png" /> with acyclic images, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020132.png" /> is compact metric ANR-space:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020133.png" /></td> </tr></table>

here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020134.png" /> is a canonical decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020135.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020136.png" /> are homomorphisms and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020137.png" /> is an isomorphism for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020138.png" /> (due to the Vietoris–Begle theorem). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020139.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020140.png" />. This result was generalized by many authors (see [[#References|[a9]]], [[#References|[a10]]], [[#References|[a4]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020141.png" /> like the degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020142.png" /> or the Kronecker characteristics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020143.png" /> some definitions are needed. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020144.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020145.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020146.png" /> be separable topological spaces (cf. also [[Separable space|Separable space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020147.png" /> be the space of compact subsets, and suppose the set-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020148.png" /> is upper semi-continuous. Such a mapping is called

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020150.png" />-acyclic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020151.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020152.png" /> (here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020153.png" /> is the set of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020154.png" /> at which the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020155.png" />-acyclicity of the images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020156.png" /> is broken);

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020158.png" />-acyclic if it is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020159.png" />-acyclic; this is equivalent to acyclicity of every image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020160.png" />. A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020161.png" /> is called generally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020163.png" />-acyclic if there exist a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020164.png" /> and single-valued continuous mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020165.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020166.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020167.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020168.png" />-Vietoris and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020169.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020170.png" />. The collection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020171.png" /> is then said to be a representation of the set-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020172.png" />, the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020173.png" /> is called a selecting pair, and the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020174.png" /> is called a selector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020175.png" />. For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020176.png" />-acyclic mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020177.png" />, the projections of the graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020178.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020179.png" /> give a selecting pair:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020180.png" /></td> </tr></table>

As an example, consider the main construction of the degree of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020181.png" /> from the unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020182.png" /> in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020183.png" /> under the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020184.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020185.png" />, is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020186.png" />-acyclic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020187.png" />. A generalization of the Vietoris–Begle theorem given by E.G. Sklyarenko ensures the existence of cohomology isomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020188.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020189.png" /> over the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020190.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020191.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020192.png" /> is given by the equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020193.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020194.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020195.png" />, is a generator of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020196.png" />, respectively <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020197.png" />, which is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020198.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020199.png" /> (a construction given by D.G. Bourgin, L. Górniewicz, and others, see [[#References|[a9]]], [[#References|[a10]]], [[#References|[a4]]]). If the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020200.png" /> under consideration is generally acyclic, then for every selecting pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020201.png" /> the set-valued mapping (the selector of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020202.png" />) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020203.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020204.png" />-acyclic, and for it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020206.png" />; applying the previous construction for the selector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020207.png" /> of the set-valued mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020208.png" />, one obtains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020209.png" /> for any selector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020210.png" />. The generalized degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020211.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020212.png" /> generated by all selecting pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020213.png" /> for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020214.png" />-mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020215.png" />. A more general construction (without the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020216.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020217.png" />-acyclic) was introduced by B.D. Gelman (see [[#References|[a10]]]); namely, the topological characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020218.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020219.png" /> is defined by the equality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020220.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020221.png" /> is a generator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020222.png" />, all the generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020223.png" /> are in accordance with the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020224.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020225.png" /> is a connecting homomorphism.

Note that an earlier definition of rotation of a set-valued field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020226.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v120/v120020/v120020227.png" />, with non-acyclic images was given in [[#References|[a8]]], [[#References|[a13]]], [[#References|[a14]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Eilenberg,   N. Steenrod,   "Foundations of algebraic topology" , Princeton Univ. Press  (1952)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.H. Spanier,   "Algebraic topology" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  E.G. Begle,   "The Vietoris mappings theorem for bicompact spaces"  ''Ann. of Math.'' , '''51''' :  2  (1950)  pp. 534–550</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Górniewicz,   "Homological methods in fixed-point theory of multi-valued maps"  ''Dissert. Math.'' , '''CXXIX'''  (1976)  pp. 1–71</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  E.G. Sklyarenko,   "Of some applications of theory of bundles in general topology"  ''Uspekhi Mat. Nauk'' , '''19''' :  6  (1964)  pp. 47–70  (In Russian)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Eilenberg,   D. Montgomery,   "Fixed point theorems for multi-valued transformations"  ''Amer. J. Math.'' , '''68'''  (1946)  pp. 214–222</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  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)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  Yu.G. Borisovich,   B.D. Gelman,   A.D. Myshkis,   V.V. Obukhovskii,   "Multivalued mappings"  ''J. Soviet Math.'' , '''24'''  (1984)  pp. 719–791  (In Russian)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  Yu.G. Borisovich,   N.M. Bliznyakov,   T.N. Fomenko,   Y.A. Izrailevich,   "Introduction to differential and algebraic topology" , Kluwer Acad. Publ.  (1995)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  L. Górniewicz,   "On non-acyclic multi-valued mappings of subsets of Euclidean spaces"  ''Bull. Acad. Polon. Sci.'' , '''20''' :  5  (1972)  pp. 379–385</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  D.G. Bouvgin,   "Cones and Vietoris–Begle type theorems"  ''Trans. Amer. Math. Soc.'' , '''174'''  (1972)  pp. 155–183</TD></TR></table>