Title: The transcendence degree of the mod p cohomology of finite Postnikov
systems
Author: Jesper Grodal
University of Copenhagen
jg@math.ku.dk
Abstract: We examine the transcendence degree of the mod p cohomology of
a finite Postnikov system E. We prove that, under mild assumptions on E,
the transcendence degree of H^*(E;F_p) is always positive, and give a
complete classification of the Postnikov systems where the transcendence
degree of H^*(E;F_p) is finite. More precisely we prove that H^*(E;F_p) is
of finite transcendence degree iff E is F_p-equivalent to the classifying
space of a p-toral group.
To obtain the results we establish a general formula for determining the
transcendence degree of an unstable algebra given in terms of the growth of
certain 'unstable Betti numbers'. This formula is easily applicable and has
for instance Quillen's theorem about the Krull dimension of the mod p
cohomology ring of a finite group as an immediate consequence.
As an application of these results we derive statements about the
n-connected cover X of a finite complex X. We show for instance
that, under suitable connectivity assumptions on X, the LS category of
X is always infinite assuming X \neq X. Finally we discuss
generalizations of the obtained results to polyGEMs.