Manifolds, homotopy theory, and related topics

In contrast to the Chas-Sullivan product, Naef showed that the Goresky-Hingston coproduct is surprisingly not a homotopy invariant of a manifold: string topology ``sees'' more than just the homotopy type. We classify the failure for the coproduct to be a homotopy invariant. Similar answers have also been obtained in recent work of Naef-Safronov and Wahl, in terms of slightly different invariants.

Along the way I'll discuss string topology of manifolds with boundary, and how it relates to the Thom isomorphism and spaces of h-cobordisms.

This is based on joint work with Lea Kenigsberg.

In this talk I'll discuss a construction on filtered spaces which implements an operation similar to a "page-turning" operation in a spectral sequence. When applied to maps between filtered objects, this implements a construction which turns categories of E_1-pages into categories of E_2-pages. I'll also discuss some ongoing work connecting this to obstruction theory and the higher Dold-Kan correspondence.

In this talk, I will discuss a cork theorem for diffeomorphisms of simply connected 4-manifolds, showing that certain diffeomorphisms can be localized to a contractible submanifold. I will outline the proof and describe some applications. This is joint work with Slava Krushkal, Anubhav Mukherjee, and Mark Powell.

We show that the topological mapping class group of a closed, 2-connected smooth manifold of dimension at least 6 is residually finite, in contrast to the situation for its smooth analogue. This implies that the topological mapping class group of such manifolds is an arithmetic group. The proof goes through a combination of embedding calculus and profinite homotopy theory, and we will focus on the technicalities of the proof.

Let \Sigma_{g,n} be an orientable surface of genus g with n punctures. The mapping class group Mod_{g,n} of \Sigma_{g,n} acts on the set of conjugacy classes of r-dimensional complex representations of \pi_1(\Sigma_{g,n}), and much effort has been expended attempting to understand the dynamics of this action, by Cantat, Eskin, Goldman, Loray, Previte, Xia and many others. I will survey recent work, joint with Landesman, classifying all finite orbits of this action when g\geq r^2; they are exactly the representations with finite image. Time permitting, I will also say something about the situation when r=2, which is much more complicated but is now completely understood by combining work of Biswas-Gupta-Mj-Whang (when g\geq 1), Bronstein-Maret, and Lam, Landesman and myself. The proofs are largely algebro-geometric in nature, taking input from (non-Abelian) Hodge theory and the Langlands program.

It is a classical fact of rational homotopy theory that the E_infinity-algebra of rational cochains on a sphere is formal, i.e., quasi-isomorphic to the cohomology of the sphere. In other words, this algebra is square-zero. This statement fails with integer or mod p coefficients. We show, however, that the cochains of the n-sphere are still E_n-trivial with coefficients in arbitrary cohomology theories. This is a consequence of a more general statement on (iterated) loops and suspensions of E_n-algebras, closely related to Koszul duality for the E_n-operads. We will also see that these results are essentially sharp: if the R-valued cochains of S^n have square-zero E_{n+1}-structure (for some rather general ring spectrum R), then R must be rational. This is joint work with Markus Land.

In joint work with Brown, Chan, and Payne, we describe a bigraded cocommutative Hopf algebra structure on the weight zero compactly supported rational cohomology of the moduli space of principally polarized abelian varieties, and use it to give lower bounds on the dimensions of these cohomology groups.  One step is to construct a coproduct on Quillen's spectral sequence, abutting to the rational homology of the one-fold delooping of the algebraic K-theory space of the integers, making it a spectral sequence of Hopf algebras.  We also relate this spectral sequence to one involving Kontsevich's graph complexes.

Function spaces have been a fundamental object of study for decades across widely varying fields of mathematics. In this talk, I’ll discuss a recipe that describes the space of functions (continuous/algebraic) between two (smooth, projective) algebraic varieties as derived indecomposables of a certain module over a graded commutative monoid in some suitably constructed symmetric monoidal category. Among other things, our result can be interpreted as an (further) algebrization (in the sense of homotopical algebra) of Bendersky-Gitler’s results from 1991, which essentially added up to saying that the space of continuous maps between two nice topological spaces has a ‘configuration space model’.  

In previous work (joint with Dustin Clausen), we introduced the reductive Borel-Serre categories: for a given ring A and any finitely generated projective module M, we associate a 1-category RBS(M) whose geometric realisation functions as a model for unstable algebraic K-theory. The RBS(M)'s assemble to a monoidal category M(A) whose realisation group completes to the K-theory space K(A), giving a sense in which the RBS(M)'s stabilise to K(A).

We have been curious about exploring other questions of stability, e.g. homological stability of the RBS(M)'s. Unfortunately, we have been unable to plug into any standard tools as these generally require some kind of homotopy commutativity and M(A) is only monoidal, not symmetric, not even braided, so its realisation is a priori only an E1-space.

In recent work, we overcome this difficulty by showing that |M(A)| naturally admits an E-infinity structure! This completely opens up the investigation of stability. In this talk, we will present the main ideas behind this proof and also explore a natural rank filtration of M(A).

The zeroeth complex topological K-theory of a space is highly computable, and encodes complex vector bundles up to stabilization. However, bridging the gap between K-theory and actual bundle theory is challenging, even for the simplest finite CW complexes.

I will discuss joint work with Hood Chatham and Yang Hu to address this problem. Our main result is to compute lower bounds for the number of stably trivial rank complex rank r topological vector bundles on complex projective n-space, for infinitely many n and r. Our methods combine Weiss calculus with computational techniques from chromatic homotopy theory. I will explain why the particular chromatic theories that we use – higher real K-theories – are particularly appropriate for this problem, and I will survey some of the computational tools at our disposal. I will also discuss some future directions and indicate where challenges arise in extending our methods.

In this paper we show an additivity result for equivariant cobordism categories, à la Steimle. Through the scanning map, it corresponds to the classical isotropy separation sequence of for genuine G-spectra. This first recovers the computation of the homotopy type of the equivariant cobordism category by Galatius/Szucs. From the proof, we also derive a fiber sequence of moduli spaces that enables us to describe the homology of the BDiff^G(M), for a compact G-manifold M verifying certain properties, in a range of degrees.

Published in 1986, Quinn provided a proof that for a simply-connected 4-manifold, any homeomorphism pseudo-isotopic to the identity is actually isotopic to the identity. Additionally for smooth manifolds, diffeomorphisms that are pseudo-isotopic are smoothly stably isotopic. Both of these results are fundamental to the world of 4-manifolds. Part of the strategy Quinn's employs for deriving these results revolves reducing the problem down to understanding and manipulating a specific collection of embedded disks. There is one particular "move" that is critical to Quinn's argument for both the smooth stably isotopic and topological pseudo-isotopy implies isotopy: the replacement criterion. However, the justification for using this move is incorrect. I plan to discuss the replacement criterion and give a way to circumvent it in order to complete Quinn's proofs.

We recently described a new geometric model for the classifying space of diffeomorphisms of 3-dimensional handlebodies as a specific open locus of the moduli space of Riemann surfaces. In particular combining our model with the work of Chan-Galatius-Payne on the hairy graph complex and on the tropical moduli space of curves we obtain a large number of unstable homology classes for this classifying space.

I will explain the main ideas that go toward proving our results, and depending on the remaining time I will also explain at the end how our model allows us to state a conjectural generalization of the isomorphism of Chan-Galatius-Payne to non-trivial local system coefficients.

This work is joint with Dan Petersen.

For any fiber bundle with compact smooth manifold fiber X ⟶ Y, Becker and Gottlieb have defined a "wrong way" map S[Y] ⟶ S[X] at the level of homology with coefficients in the sphere spectrum. Later on, these wrong way maps have been defined more generally for continuous functions whose homotopy fibers are finitely dominated, and have been since referred to as the Becker-Gottlieb transfers. It has been a long standing open question whether these transfers behave well under composition, i.e. if they can be used to equip homology with a contravariant functoriality. 

In this talk, we will approach the transfers from the perspective of sheaf theory. We will recall the notion of a locally contractible geometric morphism, and then define a Becker-Gottlieb transfer associated to any proper, locally contractible map between locally contractible and locally compact Hausdorff spaces. We will then use techniques coming from recent work of Efimov on localizing invariants and dualizable stable infinity-categories to construct fully functorial "categorified transfers". Functoriality of the Becker-Gottlieb transfers is then obtained by applying topological Hochschild homology to the categorified transfers. 

If time permits, we will also explain how one can use similar methods to extend the Dwyer-Weiss-Williams index theorem for compact topological manifolds fiber bundles to proper locally contractible maps. In particular, this shows that the homotopy fibers of a proper locally contractible map are homotopy equivalent to finite CW-complexes. Therefore, it is still unclear whether functoriality of the transfers associated to maps with finitely dominated homotopy fibers holds. 

This is a joint work with Maxime Ramzi and Sebastian Wolf.

Kirby-Siebenmann showed that, given a closed manifold M, there are only finitely many smooth manifolds, up to diffeomorphism, which are homeomorphic to M. This is not true equivariantly. Using computations of Ewing, Schultz has shown that, for certain primes p and sufficiently large n, S^{2n} can be given a smooth Z/p-action with infinitely many Z/p-smoothings. I will explain this example and give a generalization to other manifolds.