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Ponto, Kate

Relative fixed point theory. (English) Zbl 1218.55001

Algebr. Geom. Topol. 11, No. 2, 839-886 (2011).
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 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 [J. R. Klein and 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)].
Reviewer: Xuezhi Zhao (Beijing)

Cited in 7 Documents

MSC:

55M20 Fixed points and coincidences in algebraic topology
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
55P25 Spanier-Whitehead duality

Keywords:

Reidemeister trace; Nielsen number; fixed point; Lefschetz number; bicategory

Citations:

Zbl 0553.55001; Zbl 1132.57024; Zbl 1207.18001
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\textit{K. Ponto}, Algebr. Geom. Topol. 11, No. 2, 839--886 (2011; Zbl 1218.55001)
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References:

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