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.

arXiv comment9 pages, 2 figures

References in corpus (4)

• The McKean-Singer Formula in Graph Theory
• On Fredholm determinants in topology
• The strong ring of simplicial complexes
• Sphere geometry and invariants

Cited by in corpus (9)

• The amazing world of simplicial complexes
• An Elementary Dyadic Riemann Hypothesis
• Energized simplicial complexes
• A parametrized Poincare-Hopf Theorem and Clique Cardinalities of graphs
• Listening to the cohomology of graphs
• The counting matrix of a simplicial complex
• The hydrogen identity for Laplacians
• Graph complements of circular graphs
• Division algebra valued energized simplicial complexes