By J. C. Alexander, P. M. Fitzpatrick (auth.), Edward Fadell, Gilles Fournier (eds.)

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**Additional resources for Fixed Point Theory: Proceedings of a Conference Held at Sherbrooke, Québec, Canada, June 2–21, 1980**

**Example text**

O f t h e b u n d l e map 1(O). )] C-essential For each Fix(~t ) is ~ is detected S 1 x {eit@}. S1 x I. Thus This essential by a n o n - z e r o Dold ( t r a n s f e r ) index ~* (stable cohomotopy), I(~) is the element st (See Dold [4] for details of the calculation of this In this section we show that a non-zero index implies the existence of fixed points. , fiber-preserving), In this case, there of the fixed point transfer will be set in neighborhood retracts if E i U ~ n • B, such that r ~E B (i,r maps over for some plFix(f) exists p : B ~ (0,~) over n.

A Note on Abstract Integration, Trans. Amer. Math. Soc. 99 (1961), 292-297. : An introduction to Abstract Harmonic Analysis, Van Nostrand, New York, (1953). : A Fixed Point Property and Unbounded Sets in Hilbert Space, 43 Trans. Amer. Math. Soc. 258 (1980), 531-537. : Almost Convergence and Nonlinear Ergodic Theorems, J. Approxima- tion Theory, 24 (1978), 269-272. : Fourier Analysis on Groups, Interscience, New York, (1967). ON THE PRODUCT THEOREM FOR THE FIXED POINT INDEX BY ROBERTF, BR[~q Department of Mathematics University of California Los Angeles 1, INTRODUCTION of T, For U of T:X ~ X we denote by x0 CA90024 a map of a finite polyhedron i(X,T,xo) the local n Fix(T) such that = {x0}.

THEOREM (Continuation Condition for Multi-Valued Maps). Given an acyclic map be a homotopy of r ~ : M ~ N, such that and P, a closed set in 9-1(p) n 8(Mx I) = 4. If N, let 9 : Mx I ~ N 9. (N,N- P) is non-zero (or the corresponding cohomological condition), then some component of ~-l(p) Proo~ connects and M x {1}. (Adapted from the Alexander-Yorke proof for single-valued maps [ 2 ] ) . Suppose M x I, M x {0} where @-l(p) is the disjoint union of two closed subsets Q0 N (M x {0]) = @ inclusion of M as M x {i}, for and Q1 R (M x {l}): ~.