University of Sydney

Algebra Seminar

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### 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--> sigma1-sigma2+...±sigmak 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 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.