All talks will take place in the Synge Lecture Theatre
The purpose of these talks is to give two independent proofs that the Euler characteristic is multiplicative for fibrations with finitely dominated base and fibers. Remarkably, at this level of generality the result appears to be new (and is joint work with Cary Malkiewich and Maxime Ramzi). I will relate this problem to that of the problem of establishing functoriality of the Becker-Gottlieb transfer for composites of fibrations.
The first talk will provide background and give a proof of multiplicativity of the Euler characteristic using techniques available in the 1960s.
The second talk will use much newer technology involving the trace map from Waldhausen K-theory to THH.
The ordered configuration spaces of Euclidean d-space assemble to an operad, called the Ed-operad, which is ubiquitous in homotopy theory and has become increasingly relevant in geometric topology, for instance through its relation to embedding calculus in the sense of Goodwillie and Weiss. For the latter, the homotopy type of the space Aut(Ed) of derived automorphisms of Ed plays an essential role, but unfortunately our understanding of Aut(Ed) is still fairly limited. I will give an overview of what one knows and what one wishes to know about Aut(Ed), in parts based on joint works with Kupers as well as with Horel and Kupers.
I will explain the construction of a Frobenius algebra structure on Rabinowitz loop homology, following joint work with Kai Cieliebak and Nancy Hingston. Depending on the audience, I will steer course between Floer theory (symplectic) and Morse theory (topological).
Rabinowitz loop homology is an invariant of Tate flavor formed by coalescence of the homology and of the cohomology of the free loop space of a closed manifold. From a symplectic perspective it is a particular case of Rabinowitz Floer homology, and it can be described as the cone of a certain Floer continuation map. Its algebraic incarnation is the singular-, or Tate-Hochschild homology studied by Rivera-Wang.
I will describe an algebraic construction that models the passage from a topological space to its free loop space, without imposing any restrictions on the underlying space. This construction is motivated by computing non-simply connected chain level string topology explicitly and directly from the data of a triangulation of a manifold. The algebraic model is as small as it can be and also reflects how constant loops sit inside the free loop space. It is also transparent enough to reveal how string topology depends on the underlying manifold.
The input of the construction is a “categorical coalgebra" over an arbitrary ring R, i.e. a notion that may be thought as Koszul (pre)dual to that of a connective dg R-category. The output is an R-chain complex equipped with a “rotation” operator. When this construction is applied to a categorical coalgebra modeling the chains of an arbitrary simplicial set X, one obtains a chain complex that is quasi-isomorphic to the singular chains on the free loop space of the geometric realization of X. In this case, the model is generated by “necklaces” of simplices in the underlying simplicial set. I will also discuss how this models reveals how a fundamental class gives rise to an explicit smooth Calabi Yau structure lifting Poincaré duality with local coefficients.
Given a Calabi-Yau structure, Hochschild homology of a dg category has a natural multiplication. I will define another kind of structure, a noncommutative Euler structure, which allows one to define a natural comultiplication. In the case of the dg category of complexes of local systems on a homotopy type, I will relate this structure to simple homotopy theory and the comultiplication to the string coproduct. This is a report on work in progress, joint with Florian Naef.
I will explain how using the notion of pre-CY structures and a certain graphical formalism, one can write down chain-level representatives for some operations on Hochschild homology that resemble those from string topology, and that we conjecture to be a new algebraic way of calculating the Chas-Sullivan product and Goresky-Hingston coproduct. It turns out that the natural tool for assembling all this data is Efimov's definition of the "categorical formal punctured neighborhood of infinity", which generalizes the category of perfect complexes near the divisor at infinity in a compactification of an algebraic variety and is related, on the other side of mirror symmetry, to Rabinowitz Fukaya categories. This work is from a joint preprint with Manuel Rivera and Zhengfang Wang. Time allowing, I will connect to Manuel’s talk from the previous day, and describe some ongoing work where we apply this formalism to a certain DG category coming from a triangulation, where the locality of the Poincaré duality structures, in the form of a pre-CY structure, and of the operations that we defined above, is made manifest.
For a manifold W and an Ed-algebra A, the factorisation homology of W with coefficients in A admits an action by the diffeomorphism group of W and we consider its homotopy quotient W[A]. For Wg,1≔ (#g Sn × Sn) \ Ḋ2n , the collection of all Wg,1[A] is a monoid by taking boundary-connected sums. We discuss its homological stability and describe its group-completion in terms of a tangential Thom spectrum and the iterated bar construction of A. We do so by identifying the above collection with an algebra over the manifold operad, establishing a splitting result for such algebras, and studying the free infinite loop space over a given (framed) Ed-algebra.
The energy functional on the based and free loop space of a compact symmetric space is perfect as was shown by Bott and Samelson as well as by Ziller. This can be shown by constructing explicit cycles which complete the relative cycles that appear in the Morse theory of the energy functional. In this talk we show how these explicit cycles can be used to study the Chas-Sullivan product as well as the string topology coproduct on compact symmetric spaces. In particular it can be seen that there are many non-nilpotent classes in the Chas-Sullivan algebra of a compact symmetric space of higher rank whose powers correspond to the iteration of closed geodesics and we show that the string topology coproduct is trivial for all compact Lie groups of higher rank.