Manifolds, homotopy theory, and related topics

Hurwitz spaces are important moduli spaces in both number theory and algebraic geometry. From my joint work with Aaron Landesman, I will explain how Hurwitz spaces associated to finite racks satisfy homological stability, as well as modules over these Hurwitz spaces associated to more general surfaces. I will moreover explain how the stable homology of such spaces can be described in terms of that of simpler ones.

Moduli spaces of manifolds are important objects in geometric topology, playing a central role in the classification of families of manifolds. Recently, Galatius and Randal-Williams provided a complete description of the homology of stable moduli spaces of even dimensional manifolds. In this talk, I will present an odd dimensional analogue and explain the key steps in the proof of this result.

I will explain my work on extending Kontsevich's construction of characteristic classes in terms of configuration space integrals to fibre bundles with more general fibres.

I will report on joint work with Florian Naef in which we construct, for a map f of spaces over a space B such that f has compact fibers, a rational model for the fiberwise transfer of fiberwise topological Hochschild homology, considered as a map of parametrized spectra over B. This is motivated by applications to moduli spaces of manifolds: in particular we can detect the vanishing of certain cohomology classes defined by Berglund, originating from a graph complex via the classifying space of homotopy automorphisms. 

The Dundas-Goodwillie-McCarhy theorem is a fundamental theorem in algebraic K-theory, relating it to "simpler" invariants, related to topological Hochschild homology. 

The key ingredient in its proof is Goodwillie's calculus of functors: the theorem ultimately comes down to comparing the Goodwillie towers of K-theory and of topological cyclic homology, TC. Classically, this analysis uses explicit models of K-theory and connectivity estimates. 

In joint work with Harpaz, Nikolaus and Saunier, we instead study the Goodwillie derivatives of K-theory from a categorical perspective and reinterpret the cyclotomic structure on THH from this perspective. We do this by generally studying the calculus of localizing invariants, and placing in the framework of trace-theories, following older ideas of Kaledin and Nikolaus.

Recent work of Randal-Williams suggests that the homological stability of an E_2-algebra over F_p (satisfying mild assumptions) is controlled by its “stability Hopf algebra” — a discrete connected graded Hopf algebra over F_p. I will explain how this framework allows us to detect certain (12,8)-periodic families of non-trivial classes in the (relative) F_2-homology of GL_n(F_2). An immediate corollary of this is the sharpness of the slope 2/3 stability result for this homology by Galatius, Kupers, and Randal-Williams. I will also discuss some computational aspects of stability Hopf algebras when p=2.

Let p be an odd prime, and let j_p be the p-complete connective image of J spectrum. Arpon Raksit and I showed that there is an equivalence of cyclotomic E_infty-rings between THH(Z_p) and a certain "Frobenius twist" of j_p. This implies that there is an equivalence of E_infty-rings TP(Z_p) = j_p^{tS^1}. Using this, we give a new perspective on the calculation of TC(Z_p) due to Bökstedt-Madsen. In particular, we show that the splitting of TC(Z_p) is compatible with the Bökstedt-Hsiang-Madsen splitting of TC(S_p).

I will explain how to compute the homotopy ring of the Tate L-theory spectrum L^t(Z) of the integers and that any stably framed manifold gives rise to a tautological class in this homotopy ring. In fact, this association is the effect on homotopy groups of the unit map of L^t(Z). In dimensions 4k+2, I will explain how to identify this tautological class with the Kervaire invariant of a stably framed manifold. Finally, I will explain how to use the above to reprove Browder’s result that the Kervaire invariant is non-zero at most at most if h_i^2 is a permanent cycle in the Adams spectral sequence for the sphere. In particular, given the results of Hill-Hopkins-Ravenel, we reprove that there are no elements of Kervaire invariant one outside dimensions 2,6,14,30,62, and 126.

Joint work with Thomas Nikolaus and Jonathan Pedersen.

TBA