Given a self-map \(f: X\to X\) on a compact manifold, the classical Lefschetz fixed point theorem says that \(f\) must have a fixed point if its Lefschetz number \(L(f)\) is non-zero. Another refined invariant \(N(f)\), called Nielsen number, gives a lower bound for the number of fixed points of the self-map \(f\). Both of these two numbers are crucial invariants in Nielsen fixed point theory. Reidemeister and Wecken defined an invariant retaining the information of the Lefschetz number and the Nielsen number which is called Reidemeister trace. It was proved that if \(f\) is a self-map on a closed smooth manifold of dimension at least three, then \(f\) is homotopic to a fixed point free map if and only if the Reidemeister trace of \(f\) vanishes. This result generalizes not only the Lefschetz fixed point theorem, but also implies its converse.
The author of this book gives some interesting categorizations of the above fixed point theorems. The Reidemeister trace is generalized as a trace in bicategories with shadows. Its functoriality implies that the new trace will unify the existing invariants related to the Reidemeister trace, such as the invariant introduced by S. Y. Husseini [Trans. Am. Math. Soc. 272, 247–274 (1982; Zbl 0507.55001)], the invariant introduced by J. R. Klein and E. B. Williams [Geom. Topol. 11, 939–977 (2007; Zbl 1132.57024)], and the invariant introduced by M. Crabb and I. James [Fibrewise homotopy theory. Springer Monographs in Mathematics. London: Springer (1998; Zbl 0905.55001)]. Moreover, this book also contains a general and formal proof of the converse part of the Lefshetz fixed point theorem and related ones: vanishing of the trace implies fixed point free up to homotopy. As the author says, the method in it is more conceptual and does not use simplicial techniques. The key point is the approach used by E. Fadell and S. Husseini [Lect. Notes Math. 886, 49–72 (1981; Zbl 0485.55002)]: to make fixed point free is equivalent to finding a lifting of the map graph \(X \ni x\mapsto (x, f(x))\in X\times X\) with respect to the inclusion map \(X\times X - \Delta \hookrightarrow X\times X\). The identification enables us to import more arguments in homotopy theory. Furthermore, a fiberwise version of fixed point theory is also included, which was mentioned by A. Dold [Invent. Math. 25, 281–297 (1974; Zbl 0284.55007)].
This book contains reviews of classical Nielsen fixed point theory, and descriptions of how to interpret fixed point theory into category theory. Some algebraic examples of bicategories with shadows are provided.