The realization space of a \Pi-algebra:
a moduli problem in algebraic topology
D. Blanc, W. G. Dwyer, and P. G. Goerss
A \PI-algebra A is a graded group with all of the algebraic structure
possessed by the homotopy groups of a pointed connected topological
space. We study the moduli space R(A) of realizations of A, which is
defined to be the disjoint union, indexed by weak equivalence classes
of CW-complexes X with \pi_*(X)=A, of the classifying space of the
monoid of self homotopy equivalences of X. Our approach amounts to a
kind of homotopical deformation theory: we obtain a tower whose
homotopy limit is R(A), in which the space at the bottom is BAut(A)
and the successive fibres are determined by \Pi-algebra cohomology.
(This cohomology is the analog for \Pi-algebras of the Hochschild
cohomology of an associative ring or the Andre-Quillen cohomology of a
commutative ring.) The main technical tool involves working with
simplicial resolutions of spaces rather than with spaces
themselves. It seems clear that the deformation theory can be applied
with little change to study other moduli questions in topology.
In the course of working out the details, we find a simple
homotopy theoretic way to identify the space that results from taking
a functor from finite sets to sets and applying it dimensionwise to a
simplicial set. This gives an easy way to reprove and generalize many
classical connectivity theorems.