University of the South Pacific, Fiji
On Modular Homology of Saturated Simplicial Complexes
Friday 27th, June 12:05-12:55pm,
The standard simplicial homology for a complex Delta
is concerned with the Z-module ZDelta and the boundary map
which assigns to the face tau the alternating sum
of the co-dimension 1 faces of tau.
This defines a homological sequence over Z
and hence over any field.
In our previous papers we started to investigate the same
module with respect to a different homomorphism.
This is the inclusion map
\partial: ZDelta --> ZDelta given by
\partial: tau --> sigma1+sigma2+...+sigmak .
Clearly, \partial^2<> 0. However, over a field
of characteristic p>0 we have \partialp=0.
One may attempt therefore to build a
generalized modular homology theory of simplicial
complexes. (A kind of homological algebra with
\partialN=0 appears to be
mentioned first by W.Mayer and Spanier in 1947.
Recent references include Kapranov, Dubois-Violette
It occurs that even for shellable simplical
complexes the modular homology does not behave
nicely. Nevertheles, among shellable complexes
a certain class is shown to have
maximal modular homology, and these are the
so-called saturated complexes.
For example, all finite Coxeter complexes and
spherical buildings are saturated.
We will show that order complexes of geometric lattices
are saturated, and that the property of a complex to be saturated
is preserved by operations of taking cones,
suspensions and by rank-selection.
Finally we shall discuss group actions
on modular homologies of saturated complexes
and their applications to the
representation theory of finite groups.