10.1186/2251-7456-6-44

The Lefschetz number of sequences of trace class curvature

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  • Nikolai Tarkhanov*1,
  • Daniel Wallenta1,
  1. Institut für Mathematik, Universität Potsdam, Potsdam, 14469, DE

Published in Issue 2012-10-04

How to Cite

Tarkhanov, N., & Wallenta, D. (2012). The Lefschetz number of sequences of trace class curvature. Mathematical Sciences, 6(1). https://doi.org/10.1186/2251-7456-6-44
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Abstract

Abstract For a sequence of Hilbert spaces and continuous linear operators, the curvature is defined to be the composition of any two consecutive operators. This is modeled on the de Rham resolution of a connection on a module over an algebra. Purpose We wish to study those sequences for which the curvature is ‘small’ at each step, e.g., belongs to a fixed operator ideal. Methods Our methods are based on combining homological algebra with the theory of Fredholm operators in Hilbert spaces. Results We elaborate the theory of Fredholm sequences and show that any Fredholm sequence of trace class curvature can be reduced to a Fredholm complex. This allows one to introduce the Lefschetz number for cochain self-mappings of Fredholm sequences of ‘small’ curvature. Conclusion Our results raise fixed point theory for Fredholm complexes of trace class curvature.

Keywords

  • Perturbed complexes,
  • Curvature,
  • Lefschetz number,
  • primary 55U05; secondary 58J10,
  • 19K56
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References

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