Relative fixed point theory
Relative fixed point theory deals with the fixed point sets of relative maps, i.e. maps of the form \(f:(X, A)\to (X,A)\). The crucial invariant is the relative Nielsen number \(N(f; X, A)\) defined by \textit{H. Schirmer} in [Pac. J. Math. 122, 459--473 (1986; Zbl 0553.55001)], which is a lower bound for the number of fixed points for all maps in the relative homotopy class of \(f\). The author of this paper defines some relative Lefschetz numbers and relative Reidemeister traces by using traces in bicategories with shadows. The author also illustrates how to compare her new invariants with existing relative Nielsen numbers. Moreover, a relative Lefschetz fixed point theorem and its inverse are given: If \((X,A)\) is a pair of compact manifolds such that \(\dim X\geq 3\) and \(\dim X - \dim A \geq 2\), then a relative map \(f: (X,A)\to (X,A)\) is relatively homotopic to a fixed point free map if and only if the relative Reidemeister trace of \(f\) is zero. The proof of this main theorem follows that in [\textit{J. R. Klein} and \textit{E. B. Williams}, Geom. Topol. 11, 939--977 (2007; Zbl 1132.57024)]. More detailed material about bicategories with shadows can be found in the author's book [Fixed point theory and trace for bicategories. Astérisque 333. Paris: Société Mathématique de France (SMF) (2010; Zbl 1207.18001)].
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