Twisted Lefschetz numbers of infra-solvmanifolds and algebraic groups

TL;DR

This paper extends the concept of Lefschetz numbers to twisted versions on infra-solvmanifolds, showing they can be computed via determinants of matrices derived from algebraic group cohomology.

Contribution

It generalizes the linearization formula for Lefschetz numbers to twisted cases on infra-solvmanifolds using algebraic group cohomology.

Findings

Twisted Lefschetz numbers equal determinants of specific matrices.

Extension of linearization formula to twisted Lefschetz numbers.

Application of algebraic group cohomology in topological fixed point theory.

Abstract

Twisted Lefschetz numbers are extensions of the ordinary Lefschetz numbers for cohomologies with values in flat bundles. As a generalization of linearization formula for the ordinary Lefschetz number of a self-map of a nilmanifold, we show that a twisted Lefschetz number of any self-map of any infra-solvmanifold is equal to the determinant ${\rm det}(I-A)$ for some matrix $A$ by using the cohomology of algebraic groups.