The group of homotopy equivalences of
products of spheres and of Lie groups
Martin Arkowitz and Jeffrey Strom
AMS Classifications
55P10, 55P60, 55S37
Dartmouth College,
Hanover, NH 03755
Martin.Arkowitz@Dartmouth.edu
Jeffrey.Strom@Dartmouth.edu
Abstract
We investigate the group E_#(X) of self homotopy
equivalences of a space X which induce the identity
homomorphism on all homotopy groups. We obtain results
on the structure of E_#(X) provided the p-localization
X_(p) of X has the homotopy type of a p-local product of
odd-dimensional spheres. In particular, we show that
E_#(X)_(p) is a semidirect product of certain homotopy groups
pi_n(X_(p)). We also show that E_#(X)_(p) has a central
series whose successive quotients are pi_n(X_(p)),
which are direct sums of homotopy groups of p-local spheres.
This leads to a determination of the order of the p-torsion
subgroup of E_#(X) and an upper bound for its p-exponent.
These results apply to any Lie group G at a regular prime
p. We derive some general properties of E_\#(G) and give
numerous explicit calculations using MAPLE.