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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.
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