Computing Betti numbers via combinatorial Laplacians

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Abstract
We use the Laplacian and power method to compute Betti numbers of sim- plicial complexes. This has a number of advantages over other methods, both in theory and in practice. It requires small storage space in many cases. It seems to run quickly in practice, but its running time depends on a ratio, , of eigenvalues which we have yet to fully understand. We numerically verify a conjecture of Bjorner, Lov asz, Vre cica, and Zivaljevi c on the chessboard complexes C(4; 6), C(5; 7), and C(5; 8). Our verication suf- fers a technical weakness, which can be overcome in various ways; we do so for C(4; 6) and C(5; 8), giving a completely rigourous (computer) proof of the con- jecture in these two cases. This brings up an interesting question in recovering an integral basis from a real basis of vectors.

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