 
Summary: FAKE THREE CONNECTED COVERINGS
OF LIE GROUPS
J. Aguad´e, C. Broto and M. Santos
1. Introduction. In [1] it was proved that for each prime p there are infinitely many fake
3connected coverings of S3
. By "fake" we mean spaces with the same mod p cohomology
than S3
3 (as algebras over the Steenrod algebra) but different pcompleted homotopy
type. After that work was completed one could wonder if the existence of such fake spaces
was a general phenomenon and, in particular, if one could use the same methods to produce
fake three connected coverings of other Lie groups beside S3
. In this paper we prove that
the results of [1] cannot be extrapolated since, indeed, there is homotopy uniqueness up
to pcompletion for 3connected coverings of several compact connected Lie groups and
pcompact groups.
If p is a regular prime for the compact connected Lie group G then S3
is a direct factor
of G at the prime p and one can trivially construct infinitely many fake G 3 out of the
fake S3
3 constructed in [1]. If p is quasiregular for G in the sense of [10] then G splits
