×

Obstructions to homotopy invariance in parametrized fixed point theory. (English) Zbl 0972.55001

Grove, Karsten (ed.) et al., Geometry and topology, Aarhus. Proceedings of the conference on geometry and topology, Aarhus, Denmark, August 10-16, 1998. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 258, 157-175 (2000).
The authors make a correction to theorem 4.5 of the article of R. Geoghegan and A. Nicas [Am. J. Math. 116, No. 2, 397-446 (1994; Zbl 0812.55001)]. A counterexample shows that the one-parameter trace \(R(F)\) of a homotopy \(F: X\times I \to X\), which was defined in that article, is not a simple homotopy invariant in general. But, it is true in a special case (§3).
The authors also give a direct proof of the subdivision invariance of \(R(F)\), which was presented as a corollary of the simple homotopy invariance (Theorem 4.5) in that article of R. Geoghegan and A. Nicas. Thus, as the authors mention, all the results in their subsequent papers are now well proved because the simple homotopy invariance of \(R(F)\) was never used later.
For the entire collection see [Zbl 0943.00055].

MSC:

55M20 Fixed points and coincidences in algebraic topology
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
19D55 \(K\)-theory and homology; cyclic homology and cohomology
PDFBibTeX XMLCite