Recall that a smooth manifold is said to be almost closed if its boundary is a homotopy sphere. In this talk, I will describe joint work with Robert Burklund that gives a homotopy-theoretic classification of high-dimensional metastably connected almost closed manifolds. (Here I say that an n-manifold is metastably connected if it is approximately (n/3)-connected.) More precisely, we show that such a manifold is determined up to diffeomorphism by its underlying stable homotopy type, its stable normal bundle, and a homotopical quadratic form refining the Poincare duality pairing. In the case of (n-1)-connected 2n-manifolds, our classification recovers that of C.T.C. Wall. When combined with earlier joint work with Robert Burklund and Jeremy Hahn, this may be used to obtain classification results for high-dimensional metastably connected closed manifolds.

In recent work with Ben Antieau and Thomas Nikolaus, we develop new methods to compute K-theory of Z/p^n and related rings, based on prismatic cohomology. This approach can be turned into an algorithm, which we implement. The same methods also allow us to prove that K-theory of Z/p^n vanishes in large enough even degrees, and to give an explicit formula for the orders in large odd degrees. In this talk, I want to give an overview over the ingredients of these computations.

I'll talk about new results on the rational cohomology of the classifying space of Diff_\partial (U_{g,1}^n), where U_{g,1}^n is the connected sum of g copies of S^n \times S^{n+1}, minus a disc, which we consider as an odd-dimensional analogue of W_{g,1}^n, the connected sum of g copies of S^n \times S^n.

We compute the rational cohomology of BDiff_\partial (U_{g,1}^n) for large g and in degrees up to n-4. The answer looks superficially similar to the even-dimensional case in the sense that the cohomology is an exterior algebra in some generalized Miller--Morita--Mumford classes, with some notable differences.

The proof relies on the classical approach to diffeomorphism groups via surgery theory and pseudoisotopy theory. A new ingredient is the result that the gluing map BDiff_\partial (D^{2n+1}) \to BDiff_\partial (U_{g,1}^n) induces the trivial map in rational homology in the concordance stable range. This relies on the work by Botvinnik-Perlmutter on moduli spaces of odd-dimensional manifolds.

This is joint work with Jens Reinhold.

Given a closed manifold M one can construct two Euler characteristics in A-theory, one coming from the Euler class of the tangent bundle and the other one from the fact that M is (homotopy equivalent to) a finite complex. The identification of the two is determined by the underlying simple homotopy type. I will explain that after passing from A-theory to Hochschild homology, this identification can be extracted from configuration spaces of (at most 2) points on M. More concretely, I will discuss a construction that extracts the Dennis trace of Reidemeister/Whitehead-torsion from configuration spaces of points. Along the way we will see how this type of structure controls a relative intersection product which has applications in string topology.

This is joint work with Pavel Safronov.

We compute the cohomology of semi-simple Lie groups with respect to (possibly infinite-dimensional) unitary representations and in degrees below the rank. Using a new Shapiro isomorphism which is based on a breakthrough result by Leuzinger-Young in geometric group theory we are able to extend the computation from the ambient Lie group to its lattices. This is based on joint work with Uri Bader.

Let R be a ring. The term unstable algebraic K-theory will refer to any (family of) anima K(R,n) built entirely out of linear algebra internal to R^n through which the canonical maps BGL_n(R) → K(R) factorise. A classical example is Quillen's plus-construction BGL_n(R)^+. Ideally, we want a model for unstable algebraic K-theory to be closer to K(R) than BGL_n(R) is in terms of its nature and properties; for example the fundamental group of the plus-construction is closer to K_1(R) than GL_n(R) is. The term unstable algebraic K-theory was used in the 1970's by Dennis and Stein in a survey of the functor K_2, and classically unstable algebraic K-theory has been used to derive many important computational results about ``stable'' algebraic K-theory.

In joint work with Dustin Clausen, we introduce a new model for unstable algebraic K-theory inspired by a detailed study of the so-called reductive Borel--Serre compactification of locally symmetric spaces. In this talk I will introduce this model in detail; I will go through the main results and calculations that we have obtained so far and shed light on the most important aspects of the proofs.

The classifying space of the surface category is equivalent to \Omega^{\infty-1} MTSO_2 by work of Galatius-Madsen-Tillmann-Weiss. I will define a filtration of this by infinite loop spaces where the g-th stage only contains contributions from moduli spaces of surfaces of genus at most g. The associated graded of this filtration can be shown to consist of free infinite loop spaces built from curve complexes. 

With rational coefficients this filtered infinite loop space yields a spectral sequence whose E_1 page contains (the dual of) all unstable homology groups of mapping class groups of closed surfaces, and which converges to the spectrum homology of MTSO_2. As a consequence, one can for example conclude that the group H_{14}(B\Gamma_5), which was shown to be non-zero by Chan-Galatius-Payne, has rank at most 2.

After setting up the filtration and explaining the spectral sequence, I will go into more detail about the tools used to identify the associated graded.

Labeled configuration spaces B_k(M;X) of a manifold M with labels in a spectrum X generalize the notion of unordered configuration spaces B_k(M)=B_k(M;S^0). Knudsen identified labeled configuration spaces in M with the topological Quillen objects of certain spectral Lie algebras. This allows us to extract information about the mod p homology of B_k(M;X) using a bar spectral sequence and power operations on spectral Lie algebras, following the work of Knudsen on the rational homology of B_k(M;X) and Brantner-Hahn-Knudsen on their Morava E-theory. In this talk, I will explain how to compute the E^2-page of this bar spectral sequence via a May spectral sequence when p=2. Time permitting, I will talk about ongoing work with Andrew Senger on identifying the higher differentials via one-parameter deformation of comonads.

Given two closed h-cobordant manifolds M and M', how different can the homotopy types of the diffeomorphism groups Diff(M) and Diff(M') be? We will see that the homotopy groups of these two spaces are the same “up to extensions” in positive degrees in a range. Contrasting this fact, I will present an example of h-cobordant manifolds with different mapping class groups. In doing so, I will introduce a moduli space of “h-block” bundles and understand its difference with the moduli space of ordinary block bundles.

In a famous paper, Sullivan showed that the rational homotopy theory of finite type nilpotent spaces can be encoded in a fully faithful manner by mapping it to the homotopy category of commutative differential graded algebras over the rational numbers. For integral homotopy theory, a result of Mandell shows that it is faithfully captured by the integral cochains equipped with their E-infinity structute. This functor is however not full. I will explain a way of fixing this problem inspired by work of Toën, using cosimplicial binomial rings instead of E-infty differential graded algebras.

Using Seiberg-Witten theory, we construct a sequence of cohomology classes for the moduli spaces of smooth 4-manifolds. The corresponding characteristic class can detect the subtle difference between the topological category and the smooth category. And we use them to prove that homological stability fails for the moduli space of any simply-connected closed smooth 4-manifold in any degree of homology, unlike what happens in all even dimensions not equal to 4. Time permitting, we will also discuss how this characteristic class can detect an infinite-rank summand in the fundamental group of  the diffeomorphism group of many 4-manifolds (e.g. all simply-connected elliptic surfaces).

(joint with M. Anel, E. Finster and A. Joyal) 

Given a higher topos E and and a left exact localization L we construct a tower of left exact localizations $(P_n)_{n\ge 0}$ of E such that $P_0=L$. We call it the "generalized Goodwillie tower of the pair (E,L)". The construction comes out of a careful study of the factorization systems (modalities) associated to these localizations. The higher stages are described in terms of pushout product powers of its zeroth level. The layers of the tower are stable in the sense that cartesian and cocartesian squares coincide. This is a consequence of a Blakers-Massey-type theorem that comes with each generalized Goodwillie tower. Special cases of the tower are the classical Goodwillie tower, as well as Weiss' orthogonal tower.