Homology, Homotopy and Applications

Volume 17 (2015)

Number 1

Kirchhoff’s theorems in higher dimensions and Reidemeister torsion

Pages: 165 – 189

DOI: http://dx.doi.org/10.4310/HHA.2015.v17.n1.a8

Authors

Michael J. Catanzaro (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)

Vladimir Y. Chernyak (Department of Chemistry, Wayne State University, Detroit, Michigan, U.S.A.)

John R. Klein (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A.)

Abstract

Using ideas from algebraic topology and statistical mechanics, we generalize Kirchhoff’s network and matrix-tree theorems to finite CW complexes of arbitrary dimension. As an application, we give a formula expressing Reidemeister torsion as an enumeration of higher dimensional spanning trees.

Keywords

CW complex, Reidemeister torsion, combinatorial Laplacian, Kirchhoff’s formulae

2010 Mathematics Subject Classification

Primary 55U15, 57M15, 57Q10. Secondary 05C05, 05C21, 05E45, 82C31.

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