### Valeriy Mnukhin

University of the South Pacific, Fiji

### On Modular Homology of Saturated Simplicial Complexes

**Friday 27th, June 12:05-12:55pm,
Stephen Roberts. **
The standard simplicial homology for a complex *Delta*
is concerned with the *Z*-module *ZDelta* and the boundary map
*
tau--> sigma*_{1}-sigma_{2}+...±sigma_{k}
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 --> sigma*_{1}+sigma_{2}+...+sigma_{k} .
Clearly, *\partial^2<> 0*. However, over a field
of characteristic *p>0* we have *\partial*^{p}=0.
One may attempt therefore to build a
generalized *modular homology theory* of simplicial
complexes. (A kind of homological algebra with
*\partial*^{N}=0 appears to be
mentioned first by W.Mayer and Spanier in 1947.
Recent references include Kapranov, Dubois-Violette
and Cassel.)

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.