Online seminar, this Tuesday at 10:00, U. Haifa Topology & Geometry seminar on May 12, 2020
Geometry & Topology Seminar
Speaker: Drew Heard, (Regensburg University)
Topic: Rational local systems and connected finite loop spaces
Place: This is an online seminar. Please email David Blanc <firstname.lastname@example.org> for the Zoom ID and password.
Date: Tuesday, May 12, 2020
Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group G has an algebraic model – that is, it is (Quillen) equivalent to the category of dg-modules over an abelian category. This conjecture has been verified in a number of cases. Of particular interest, are the full subcategories of free and cofree G-spectra, where the abelian categories are the derived category of dg-torsion and dg-L-complete H^*(BG)-modules respectively, at least when G is connected, as shown by Greenlees–Shipley (for free G-spectra) and Pol–Williamson (for cofree G-spectra). We take a different approach to these results, starting from the identification of cofree G-spectra as rational local systems on the classifying space of BG (equivalently, a rational spectrum parametizied over BG). We give an algebraic model for rational local systems over a connected finite loop space (X,BX), and show that when X = G we recover the (connected case of) results of Greenlees–Shipley and Pol–Williamson. Throughout, we pay careful attention to the role of torsion and complete objects in a stable category.