ON FINITE GROUPS AND HOMOTOPY THEORY

By Ran Levi

Appeared in Memoirs of the American Mathematical Society, Vol. 567 (1995)


Abstract:

Let p  be a fixed prime number. Let G  denote a finite 
p-perfect group. We study the 
homotopy type of the p-completed classifying space BG^p. 
The paper is divided into two 
parts. In part 1  we study the homology, homotopy and 
stable homotopy of \Omega 
BG^p, where G is a finite p-perfect group. We 
construct an algebraic analogue of the 
Quillen's "plus" construction for differential graded 
coalgebras. This construction is 
used to show that given a finite p-perfect group G,  
the loop spaces \Omega BG^p admits 
integral homology exponents.  We give examples to 
show that in some cases our bound 
is best possible. We show that in general BG^p 
admits infinitely many non-trivial k-
invariants and discuss some examples where homotopy 
exponents exist. Finally classical 
constructions in stable homotopy theory are used to 
show that the stable homotopy 
groups of these loop spaces also  have exponents.  

In Part 2 we define the concept of resolutions by 
fibrations over an arbitrary family of 
spaces. We construct examples for finite p-perfect 
groups G such that \Omega BG^p is 
finitely resolvable over the family of spheres and 
their iterated loop spaces. In 
particular we get such resolutions for loop spaces 
associated with most finite groups 
of Lie type and with some of the polynomial cohomology 
spaces of Ewing and Clark. 
Various sporadic examples are also discussed.