We study a class of spaces whose importance in homotopy theory was first
highlighted by work of Dold in the 1960s, and that we accordingly call Dold spaces.
These are the spaces that possess a partition of unity supported in sets
that are contractible to a point within the ambient space. Dold spaces form
a broader class than spaces homotopy equivalent to CW complexes, but
share the feature that a number of well known weak equivalences are genuine
ones if Dold spaces are involved. In this paper we give a first systematic
investigation of Dold spaces. After listing their elementary properties, we
study homotopy pullbacks involving Dold spaces and simplicial objects in the
category of Dold spaces. In particular, we show that the homotopy colimit of a
diagram of Dold spaces is a Dold space and that the topological realization
functor preserves fibration sequences if the base is a path-connected Dold
space in each dimension. It follows that the loop space functor commutes
with realization up to homotopy for Dold spaces. Finally, we give simple
conditions which assure that free algebras over a topological operad are Dold
spaces.
Keywords
free algebra over operad, homotopy pullback, homotopy
pushout, James construction, numerable cover, simplicial
spaces
