Jaka Smrekar

Compact open topology and CW homotopy type

Abstract The class W of spaces having the homotopy type of a CW complex is not closed under formation of function spaces. In 1959, Milnor proved the fundamental theorem that, given a space Y∈ W and a compact Hausdorff space X, the space YX of continuous functions X→Y, endowed with the compact open topology, belongs to W . P.J.xa0Kahn extended this in 1982, showing that Y X ∈ W if X has finite n-skeleton and πk(Y)=0, k>n. Using a different approach, we obtain a further generalization and give interesting examples of function spaces Y X ∈ W where X∈ W is not homotopy equivalent to a finite complex, and Y∈ W has infinitely many nontrivial homotopy groups. We also obtain information about some topological properties that are intimately related to CW homotopy type. As an application we solve a related problem concerning towers of fibrations between spaces of CW homotopy type.

Function spaces of CW homotopy type are Hilbert manifolds

Let X be a countable CW complex and Y an ANR (for metric spaces) and let Y X denote the space of continuous maps from X to Y with the compact-open topology. We show that, under mild restrictions, the following are equivalent: (1) Y X is an ∂ 2 -manifold, (2) Y X is an ANR, (3) Y X has the homotopy type of a CW complex. We also give a few interesting examples and applications.

Homotopy type of mapping spaces and existence of geometric exponents

Abstract Let Y be a simply connected finite complex and let p be a prime. Let Sm [p –1] denote the complex obtained from the m-sphere by inverting p. It is shown in this paper that Y has an eventual H-space exponent at p if and only if the space map*(Sm [p –1], Y) of pointed maps Sm [p –1] → Y has the homotopy type of a CW complex for some (and hence all big enough) m. This makes it possible to interpret the question of eventual H-space exponents in terms of phantom phenomena of mapping spaces.

Compact maps and quasi-finite complexes

Abstract The simplest condition characterizing quasi-finite CW complexes K is the implication X τ h K ⇒ β ( X ) τ K for all paracompact spaces X . Here are the main results of the paper: Theorem 0.1 If { K s } s ∈ S is a family of pointed quasi-finite complexes, then their wedge ⋁ s ∈ S K s is quasi-finite. Theorem 0.2 If K 1 and K 2 are quasi-finite countable CW complexes, then their join K 1 * K 2 is quasi-finite. Theorem 0.3 For every quasi-finite CW complex K there is a family { K s } s ∈ S of countable CW complexes such that ⋁ s ∈ S K s is quasi-finite and is equivalent, over the class of paracompact spaces, to K. Theorem 0.4 Two quasi-finite CW complexes K and L are equivalent over the class of paracompact spaces if and only if they are equivalent over the class of compact metric spaces. Quasi-finite CW complexes lead naturally to the concept of X τ F , where F is a family of maps between CW complexes. We generalize some well-known results of extension theory using that concept.

Homotopy Characterization of ANR Function Spaces

Let be an absolute neighbourhood retract (ANR) for the class of metric spaces and let be a topological space. Let denote the space of continuous maps from to equipped with the compact open topology. We show that if is a compactly generated Tychonoff space and is not discrete, then is an ANR for metric spaces if and only if is hemicompact and has the homotopy type of a CW complex.

Periodic homotopy and conjugacy idempotents

A self-map f on the CW complex Z is a periodic homotopy idempotent if for some r ≥ 0 and p > 0 the iterates f r and f r+P are homotopic. Geoghegan and Nicas defined the rotation index RI(f) of such a map. They proved that for r = p = 1, the homotopy idempotent f splits if and only if RI(f) = 1, while for r = 0, the index RI(f) divides p 2 . We extend this to arbitrary p and r, and generalize various results related to the splitting of homotopy idempotents on CW complexes and conjugacy idempotents on groups.

On the Hodge conjecture for q–complete manifolds

We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any

Turning a self-map into a self-fibration

q

Algebraic properties of quasi-finite complexes

-complete complex manifold

CW towers and mapping spaces

X