Title: One can hear the Euler characteristic of a simplicial complex,
Abstract: We prove that that the number p of positive eigenvalues of the connection
Laplacian L of a finite abstract simplicial complex G matches the number b of
even dimensional simplices in G and that the number n of negative eigenvalues
matches the number f of odd-dimensional simplices in G. The Euler
characteristic X(G) of G therefore can be spectrally described as X(G)=p-n.
This is in contrast to the more classical Hodge Laplacian H which acts on the
same Hilbert space, where X(G) is not yet known to be accessible from the
spectrum of H. Given an ordering of G coming from a build-up as a CW complex,
every simplex x in G is now associated to a unique eigenvector of L and the
correspondence is computable. The Euler characteristic is now not only the
potential energy summing over all g(x,y) with g=L^{-1} but also agrees with a
logarithmic energy tr(log(i L)) 2/(i pi) of the spectrum of L. We also give
here examples of L-isospectral but non-isomorphic abstract finite simplicial
complexes. One example shows that we can not hear the cohomology of the
complex.