A theorem on families of acyclic sets and its applications
Antoni A. Kosinski · Pacific Journal of Mathematics · 1962
In the first part of this note we discuss a group of theorems dealing with geometric configurations arising when we assign in continuous way compact, acyclic sets to /b-planes in the Euclidean ^-dimensional space E n .A fairly representative example of those theorems is as follows:Suppose that to every (unoriented) A -plane H through a point a of E n there is upper semi-continuously assigned a compact and acyclic set Φ(H) c H. Then for some plane H Q , a e Φ(H 0 ).In fact, we will prove a much more general theorem of which the above is one of the consequences.In the second part of this note we give various applications of the above theorems.They are related to the theory of convex sets ( § 2.1-2.4),mappings of manifolds ( § 2.6), and to some relations between vector fields and involutions on S n ( §2.5).The author wishes to acknowledge his indebtedness to Dr. M. Hirsch for valuable suggestions and to Dr. J. W. Jaworowski whose generalization of the author's previous results was the starting point for the present paper.lFamilies of compact sets over Grassmannians.1.1 H n (X) will denote the nth.Cech homology group of the space X with the group Z 2 of integers mod 2 as the group of coefficients.We will say that Xis acyclic if X is connected andLet X be a compact metric space and let Φ : X -• 2 r be an upper semi-continuous mapping of X into the space 2 Y of all nonempty compact subsets of a space Y.The triple J^ = {X, Y, Φ) will be called a family [3].The set X will be called the basis of J^"", the sets Φ(x)the elements of ^ the set \J*eχ Φ(x) c Y-the field of jr.The field will be also denoted Φ(X).A family jβ?~ is said to be acyclic if all its elements are acyclic.If ^r = {χ f γ f φ) is a family then the subset M = {(x, y)\ye Φ(x)} of the cartesian product X x Y is called the graph of ^.Mis a closed subset of X x Y (and, hence, compact) because of the upper semi-continuity of Φ, and this is the only reason for requiring the upper semi-continuity of Φ. G v>q will denote the Grassmannian of (unoriented) g-planes through