Axioms for the Lefschetz number as a lattice valuation

P. Christopher Staecker

Abstract


We give new axioms for the Lefschetz number based on Hadwiger’s characterization of the Euler characteristic as the unique lattice valuation on polyhedra which takes value 1 on simplices. In the setting of maps on abstract simplicial complexes, we show that the Lefschetz number is unique with respect to a valuation axiom and an axiom specifying the value on a simplex. These axioms lead naturally to the classical computation of the Lefschetz number as a trace in homology. We then extend this approach to continuous maps of polyhedra, assuming an extra homotopy invariance axiom. We also show how this homotopy axiom can be weakened.

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How to Cite this Article:

P. Christopher Staecker, Axioms for the Lefschetz number as a lattice valuation, Advances in Fixed Point Theory, Vol 4, No 2 (2014), 149-159

Copyright © 2014 P. Christopher Staecker. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advances in Fixed Point Theory

ISSN: 1927-6303

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