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Hochschild and cyclic theory for categorical coalgebras: an algebraic model for the free loop space and its equivariant structure

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posted on 2024-04-18, 21:16 authored by Daniel C TolosaDaniel C Tolosa

We develop a cyclic theory for categorical coalgebras and show that, when applied to the categorical coalgebra of singular chains on a space, this provides an algebraic model for its free loop space as an S1-space. In other words, the natural circle action on loop spaces, given by rotation of loops, is encoded in the algebraic structure. In particular, the cyclic homology of the categorical coalgebra of singular chains on a topological space X is isomorphic to the S1-equivariant homology of the free loop space. This extends known results relating cyclic theories for the algebra of chains on the based loop space and the equivariant homology of its free loop space. In fact, our statements do not require X to be simply connected, and we work over an arbitrary commutative ring. Along the way, we introduce a family of polytopes, coined as Goodwillie polytopes, that control the combinatorics behind the relationship of the coHochschild complex of a categorical coalgebra and the Hochschild complex of its associated differential graded category.

History

Degree Type

  • Doctor of Philosophy

Department

  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Manuel Rivera

Additional Committee Member 2

Jeremy Miller

Additional Committee Member 3

Ralph Kaufmann

Additional Committee Member 4

Mahmoud Zeinalian

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