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Deformation Theory Seminar

Wednesday, August 10, 2016 - 2:00pm

Sam Smith

St Joseph U

Location

University of Pennsylvania

DRL 3C8

We discuss two open problems on self-equivalences in the context of rational homotopy theory.

The first concerns the group E(X) of homotopy classes of homotopy self- equivalences of X. Originally posed by Donald Kahn, the problem is to determine which groups G can be realized as E(X) for some space X. We discuss the recent result of Christina Costoya and Antonio Viruel proving all finite groups can be realized as E(X) for some rational space X.

The second problem concerns the monoid aut(X) of all homotopy self- equivalences of X and its classifying space Baut(X). The problem here, posed by Mike Schlessinger and Jim Stasheff, is to determine which spaces Y can be realized as Baut(X) for some space X. We discuss recent joint work with Greg Lupton on this problem in rational homotopy theory.

These topics were the subject of a course given at the Summer School on Rational Homotopy Theory in Rabat, Morocco. The conference celebrated Dennis Sullivan and Jim Stasheff on their respective birthdays.