Session 35. Topological fixed point theory and related topics

An averaging formula for Reidemeister traces

Xuezhi Zhao, Capital Normal University, China
Let \(f\colon X\to X\) be a self-map on a connected compact polyhedron. Assume that \(f\) admits a lifting with respect to an \(n\)-fold regular covering \(q\colon \bar X \to X\). It is well-known that there is an averaging formula \(L(f)= \frac{1}{n}\sum_{\bar f\in \mathrm{lift}(f, q)} L(\bar f)\) for Lefschetz numbers (see [1]). Here, \(\mathrm{lift}(f, q)\) stands for the set of all liftings of \(f\) with respect to the covering \(q\colon \bar X\to X\). Moreover, an averaging formula \(N(f)= \frac{1}{n}\sum_{\bar f\in \mathrm{lift}(f, q)} N(\bar f)\) for Nielsen numbers was obtained in [3] under some assumptions on the given self-map \(f\) or the space \(X\). In this talk, we shall show that there does exist an averaging formula for Reidemeister traces. The Reidemeister trace of a self-map is also a classical invariant containing the information of both the Lefschetz number and the Nielsen number. Such a result may illustrate the idea of [2] for all classical invariants in Nielsen fixed point theory.
References
  1. B. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math. 14, Amer. Math. Soc., Providence 1983.
  2. J. Jezierski, Nielsen number of a covering map. Fixed Point Theory Appl. 2006, Special Issue, Art. ID 37807, 11 pp.
  3. S. W. Kim, J. B. Lee, K. B. Lee, Averaging formula for Nielsen numbers. Nagoya Math. J. 178 (2005), 37 - 53.
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