Algebra/Topology Seminar
Speaker: Sune Precht Reeh (Copenhagen)
Title: Transfer for free loop spaces and covering maps
Given any finite sheeted covering map E --> B we know that the transfer in homology and cohomology can be exhibited as a map of suspension spectra \Sigma^\infty_+ B --> \Sigma^\infty_+ E.
If we consider the n-fold free loop spaces L^n E and L^n B, we can also define a transfer from L^n B to L^n E which sends a loop in B to zero if it doesn't lift to a loop in E. However, this transfer does not commute with the evaluation maps L^n(X) x (S^1)^n --> X even in very simple cases.
In a joint project with Tomer Schlank and Nathaniel Stapleton we construct a "twisted" transfer for L^n(-) x (S^1)^n that uses the S^1-factors non-trivially to twist the free loop space in such a way that the twisted transfer commutes with evaluation.
This twisted transfer feeds into an ongoing project of ours to construct Hopkins-Kuhn-Ravenel generalized character maps for abstract saturated fusion systems.