PhD defense: Massimiliano Ungheretti
Title: On the algebraic structure of Hochschild complexes and the free loop space.
The overarching themes of this thesis are the algebraic structure of Hochschild complexes and free loop spaces. It is a presentation of three projects in progress, followed by two papers.
In the first project, which is based on work of Kontsevich--Soibelman and Wahl--Westerland, we define a three-coloured differential graded operad $T$ using graph complexes, and sketch how it acts on $(C^*(A,A),C_*(A,A),A)$, the triple of Hochschild (co)chains of an $\Ainf$-algebra $A$.
The Hochschild cochains $C^*(A,A)$ are not functorial in the algebra $A$. In the second project we make sense of `natural' operations on $C^*(-,-)$ by defining a functor from multiplicative PROPs to chain complexes instead. This functor recovers the usual definition of Hochschild cochains when applied to endomorphism algebras, and its definition is based on work of McClure and Smith.
The third project discusses operations on cyclic chains. In particular, for a given operad that acts on the Hochschild chains on an algebra, we construct an operad acing on the cyclic chains of that algebra.
In the first paper we fix a gap in the paper “Cyclic homology and equivariant homology” by John D.S. Jones. To achieve this, we use the $E_\infty$-structure of singular cochains to construct a homotopy coherent map between the cyclic bar construction of the differential graded algebra of cochains on a space and a model for the cochains on its free loop space.
The second paper proves an $O(2)$-equivariant version of the Jones isomorphism, relating Borel $O(2)$-equivariant cohomology of free loop spaces to negative dihedral homology, a variation of cyclic homology. After discussing a variation of the de Rham isomorphism, we apply the results to calculate the rational Borel $O(2)$-equivariant cohomology of the free loop space of the 2-sphere.
Supervisor: Prof. Nathalie Wahl, Univerity of Copenhagen
Assessment committee:
Prof. Ryszard Nest, Math, Chairman, University of Copenhagen
Ass. Prof. Marcel Bökstedt, Aarhus University
Ass. Prof. Grégory Ginot, Université Pierre et Marie Curie