Nordic Topology Meeting 2014 - Abstracts

Abstract: The cohomology of the diffeomorphism group of an orientable surface stabilizes as the genus increases. The stable cohomology is described by the Mumford conjecture, which was proved by Madsen-Weiss.

In this talk, we consider \(2d\)-dimensional manifolds (\(2d > 4\)), where one stabilizes by taking connected sums with \(S^d \times S^d\). Unlike in the surface case, the monoid of homotopy automorphisms and the diffeomorphism group do not have homotopy equivalent classifying spaces. Still, both cohomologies stabilize. The stable cohomologies were determined recently by Berglund-Madsen and Galatius-Randal-Williams, respectively, but the precise relation between them remains to be understood. To measure the difference, one introduces the group of 'block diffeomorphisms'. I will report on a new result which describes the stable cohomology of the block diffeomorphism group in terms of the homology of certain decorated graph complexes. Curiously, closely related graph complexes were introduced by Conant-Kassabov-Vogtmann in the study of automorphisms of free groups. We are able to identify, at least conjecturally, what parts of the decorated graph complex correspond to homotopy automorphisms and diffeomorphisms, respectively. This is joint work with Ib Madsen.

Abstract: In the first part of the talk we will briefly introduce ordinary persistent homology and survey some of its applications. In the second part we shall see how applications call for more generalized theories, and discuss the mathematical challenges that have prevented such theories from yet to be applied to the sciences.

Abstract: We consider spaces of divisors on a compact, orientable surface. The points of the divisors come in a number of colors. Two points of different colors may only coincide according to certain rules. The fundamental group of the corresponding configuration space is determined. The tools for the computation will include a linking number for braids (with colored strands) on a surface, and a version of graph cohomology.

Abstract: If \(A\) is a commutative ring, the categories of commutative \(A\)-algebras and the category of commutative \(\mathit{HA}\)-algebras (where \(H\) is the Eilenberg-MacLane construction) are very different, even thought the underlying associative structure is well understood. Only in very special circumstances are we able to describe explicitly the homotopy types in the latter category. One example occurs in connection with Postnikov sections, as observed by Basterra, Kriz and Mandell.

When investigating higher topological Hochschild homology (tensoring with spheres) of rings of integers, Lindenstrauss, Richter and myself were lucky enough to be in favorable circumstances when the dimension was greater than one. This exposes an interesting schism between topological Hochschild homology and its higher counterparts, contrasting with Pirashvili’s result in characteristic zero where the difference is shown to be very minor. In this talk I will try to discuss the nature of these phenomena.

Abstract: The Segal conjecture, in its general form, identifies the mapping space \(\mathrm{map}(BG,\Omega^\infty S^\infty)\) with a certain completion of the (genuine) \(G\)-equivariant infinite loop space \(\Omega^\infty S^\infty\), for \(G\) a finite group. Its verification by G. Carlsson was one of the big triumphs of homotopy theory in the 80's. There is an interesting intermediate space between \((\Omega^\infty S^\infty)^G\) and the completion, namely \(\Omega B \coprod_n \mathrm{map}(BG,B\Sigma_n^+)\), which has so far not been understood. (The plus denotes Quillen's plus construction.) In my talk I'll give a complete description of this space, which turns out to be related to the \(p\)-fusion in the finite group. The result is interesting even on components, where it identifies the Grothendieck group of maps with a product of the (uncompleted) Burnside rings of the \(p\)-fusion systems of \(G\).

Abstract: I will discuss a construction of a higher category of \(E_n\)-algebras and iterated bimodules, generalizing the classical bicategory of algebras and bimodules. This leads to generalizations of the Picard and Brauer groups, which have been studied in stable homotopy theory as interesting invariants of ring spectra, and should also lead to an "algebraic" construction of factorization homology as an extended topological quantum field theory.

Abstract: I will report on recent joint work with Emanuele Dotto about so-called \(G\)-diagrams. Let \(G\) be a finite which acts on a small category \(I\). A \(G\)-diagram in a category \(\mathscr{C}\) is an \(I\)-indexed diagram \(X\) in \(\mathscr{C}\) along with a kind of generalized \(G\)-action on \(X\) which is compatible with the \(G\)-action on \(I\). If \(X\) is a \(G\)-diagram of spaces then the homotopy limit and homotopy colimit of the underlying diagram inherit natural \(G\)-actions from the structure on \(X\). This can be used to construct equivariant suspension and loop spaces with respect to nice representations of \(G\). We provide a model structure on the category of \(G\)-diagrams in a nice model category such that these constructions are equivariantly homotopy invariant. Goodwillie's notion of linearity for homotopy functors generalizes to this equivariant context using cubical \(G\)-diagrams. This theory leads to a new proof of the Wirthmüller isomorphism theorem for orthogonal \(G\)-spectra.

Abstract: In recent years there have been a lot of cases where people rediscover, and/or misunderstand the surgery exact sequence. It does not help that the version in Wall's book is wrong. The talk will discuss this phenomenon.

Abstract: Calculating the residues for rational integrals in complex variables is a classical problem in mathematics. It is directly related to questions on algebraic cycles, their cohomology classes, and the Abel-Jacobi map. In this talk I will present joint work with Michael Hopkins in which we use topological cohomology theories to shed some new light on the Abel-Jacobi map.

Abstract: This is joint work with Lück, Reich and Varisco. We use a variant of the cyclotomic trace map to topological cyclic homology to detect elements in \(K_n(\mathbb{Z}G) \otimes \mathbb{Q}\), the rationalized higher algebraic \(K\)-groups of an integral group ring. The aim is to generalize a theorem of Bökstedt-Hsiang-Madsen from the case of torsion-free groups \(G\) to more general discrete groups \(G\), and we achieve this assuming a generalization of the Leopoldt conjecture due to Schneider.

Nordic Topology Meeting 2014