Vol. 21, No. 3, 1967

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Fixed points and fibre

Robert F. Brown

Vol. 21 (1967), No. 3, 465–472

Let = (E,p,B) be a (Hurewicz) fibre space and let λ be a lifting function for For W a subset of B, a map f : p1(W) E is called a fibre map if p(e) = p(e) implies p(f(e)) = p(f(e)). Define f : W B to be the map such that fp = pf. If [W f(W)]V B where V is pathwise connected, define fbV : p1(b) p1(b), for b W, by fbV (e) = λ(f(e))(1) where ω;I V is a path such that ω(0) = f(b)and ω(1) = b. Let i be a fixed point index defined on the category of compact ANR’s and let Q denote the rationals. The main result of this paper is: THEOREM 1. Let = (E,p,B) be a fibre space such that E,B, and all the fibres are compact ANR’s. Let f : E E be a fibre map. If U is an open subset of B such that f(b)b for all b bd(U) and cl [U f(U)] V where V is open and pathwise connected and ℱ|V = (p1(V ),p,V ) is Q-orientable, then

i(f,p−1(V )) = i(f− ,U ).L(fVb )

where L(fbV ) is the Lefschelz number of fbV for any b U.

Mathematical Subject Classification
Primary: 55.50
Received: 15 June 1965
Published: 1 June 1967
Robert F. Brown
Department of Mathematics
University of California, Los Angeles
Los Angeles CA 90095-1555
United States