Enriched homology and cohomology modules of simplicial complexes

Details Enriched homology and cohomology modules of simplicial complexes Enriched homology and cohomology modules of simplicial complexes

2004-11-25

arxiv.org/abs/math/0411570v2

J. Algebraic Combinatorics 25 (2007), no. 3, p. 285-307

Mathematics

Combinatorics

22 pages, some new material on designs

Scientific paper

10.1007/s10801-006-0038-z

For a simplicial complex X on {1,2, ..., n} we define enriched homology and cohomology modules. They are graded modules over k[x_1, ..., x_n] whose ranks are equal to the dimensions of the reduced homology and cohomology groups. We characterize Cohen-Macaulay, l-Cohen-Macaulay, Buchsbaum, and Gorenstein* complexes X, and also orientable manifolds in terms of the enriched modules. We introduce the notion of girth for simplicial complexes and make a conjecture relating the girth to invariants of the simplicial complex. We also put strong vanishing conditions on the enriched homology modules and describe the simplicial complexes we then get. They are block designs and include Steiner systems S(c,d,n) and cyclic polytopes of even dimension.

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