20/03/2025, 15:00 — 16:00 —
Room P3.10, Mathematics Building
Manuel Rivera, Purdue University
An algebraic model for the free loop space
I will discuss geometric, algebraic, and combinatorial constructions related to loop spaces in topology. The talk will revolve around a functorial construction that models the passage from a topological space X to its free loop space LX. The input is a coalgebra equipped with additional structure and the output is a chain complex with a compatible “rotation” operator. The construction is dual in an appropriate sense to the Hochschild complex of a dg algebra/category. When applied to the coalgebra of chains on X, suitably interpreted, it produces a chain complex that is naturally quasi-isomorphic to the chains on LX with rotation operator corresponding to the circle action. This statement does not require any hypotheses on X (such as simple connectivity, nilpotence, finite type, etc…) or on the underlying ring of coefficients. The model turns out to be useful when studying and computing explicitly the structure of the free loop space of a manifold.
