You are cordially invited to the Topology Seminar organized by the Department of Mathematics.
Speaker: Basak Kucuk (University of Göttingen)
“The Conjecture of Klein and Williams for the Equivariant Fixed Point Problem”
Abstract: Klein and Williams developed an obstruction theory for the homotopical equivariant fixed point problem, which asks whether an equivariant map can be deformed, through an equivariant homotopy, to a map with no fixed points [KW, Theorem H]. An alternative approach was given by Fadell and Wong [FW], using a collection of Nielsen numbers. The Nielsen number is a finer invariant than the Lefschetz number in the sense that it provides a converse to the Lefschetz fixed point theorem. Klein and Williams [KW] conjectured that these Nielsen numbers could be computed from their invariant.
In this talk, we present our findings on this conjecture by providing an explicit decomposition of the Klein–Williams invariant under the tom Dieck splitting. We further discuss the application of the equivariant fixed point problem to the periodic point problem of period n. In this setting, we show that the Klein–Williams invariant and the Nielsen numbers N(fk), for all k dividing n, carry the same amount of information. However, they are not exactly the same invariants, and if time permits, we will conclude with an explicit example illustrating this difference.
References:
[KW] J. R. Klein and B. Williams, Homotopical intersection theory II, Math. Z. 264 (2010). [FW] E. Fadell and P. Wong, On deforming G-maps to be fixed point free, Pacific Journal of Mathematics 132 (1988).
Date: July 11, Friday, 2025
Time: 13:30 UTC+3
Place: ZOOM
To request the event link, please send a message to cihan.okay@bilkent.edu.tr