Vol. 95, No. 1, 1981

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Splitting homotopy idempotents which have essential fixed points

Ross Geoghegan

Vol. 95 (1981), No. 1, 95–103

We consider the problem of whether every homotopy idempotent f : X X, X being a finite CW complex, splits as a domination by X of some CW complex Y . This problem has a history (which we explain) both in abstract homotopy theory and in geometric topology. If f is a pointed homotopy idempotent, it is known that f splits; if X is permitted to be infinite-dimensional, it is known that f need not split; and the obstruction to splitting is describable entirely in f : π1(X,x) π1(X,x). The difficulty, then, lies in requiring X to be finite and permitting f to be merely freely homotopic to f2.

Our idea is to compare the fixed point theory of f (this is where we use the fact that X is finite) with its homotopy theory. We apply a theorem about the fundamental-group behavior of a homotopy idempotent which has essential fixed points, which we proved in the preceding paper. We believe that this theorem may eventually be used to prove our conjecture that f splits when the Lefschetz number L(f) is nonzero. In the present paper we only succeed in getting part of the way to such a result, by showing (Theorems 1.16 and 1.17) just how subtle a counter-example to the conjecture would have to be.

The problem of whether every homotopy idempotent on a finite complex splits is equivalent to the well-known problem in shape theory of whether every FANR is a pointed FANR (equivalently: does a compactum shape dominated by a complex have the shape of a complex?) In those terms, we are looking at the case of FANR’s whose Čech Euler characteristic is nonzero.

Mathematical Subject Classification 2000
Primary: 55P99
Secondary: 55M20
Received: 13 November 1979
Published: 1 July 1981
Ross Geoghegan