CW Type of Inverse Limits and Function Spaces

We continue the investigation of CW homotopy type of spaces of continuous functions between two CW complexes begun by J. Milnor in 1959 and P. Kahn in 1984. Viewing function spaces as particular cases of inverse limits we also study certain inverse systems of fibrations between CW homotopy type spaces. If the limit space Z∞ of an inverse sequence {Zi} of fibrations between CW type spaces has CW type then a subsequence of {ΩZi} splits into the product of a sequence of homotopy equivalences and one of nullhomotopic maps. If for some N > 0, all spaces Zi have πk(Zi) = 0 for k > N , then the question of CW type of Z∞ depends solely on the induced functions πk(Zj) → πk(Zi). This applies to Zi = Y Li where πk(Y ) = 0 for k > N and {Li} is an ascending sequence of finite complexes. Here Z∞ = Y ∪Li , the space of continuous functions (∪Li) → Y with the compact open topology. In general, if the path component of g ∈ Y X has CW type then Ω(Y X , g) → Ω(Y , g|L) is a homotopy equivalence for a countable subcomplex L of X. A suitable converse holds as well. Function spaces of CW type lack phantom phenomena in a strong sense. This provides interesting examples. One is a space of pointed maps that is weakly contractible but not contractible. Next, for X the localization of a finite complex at a set of primes P , the question of CW type of Y X is related, and sometimes equivalent, to that of eventual geometric H-space exponents of Y . If Y is a P -local and X a simply connected complex then localization X → X(P ) induces a genuine homotopy equivalence Y X(P) → Y X regardless of whether Y X has CW type or not. For Y = K(G,n) we give necessary and sufficient conditions for Y X to have CW type in terms of homology of X. If ⊕nπn(Y ) is finitely generated and X is 1-connected then we give necessary and ‘almost’ sufficient conditions. Some properties of CW complexes X are equivalent to Y X having CW homotopy type for a certain family of complexes Y . For example, X is finitely dominated if and only if π1(X) is finitely presented and Y X has CW type for all complexes Y .