Fall 2024 Algebraic topology seminar

Abstract: Understanding the algebraic K-theory of the sphere spectrum has long been recognized as a fundamental problem in algebraic and differential topology. Since the homotopy fiber of its p-completed cyclotomic trace depends only on the zeroth homotopy group, we can apply algebraic methods to study it. Blumberg and Mandell’s work demonstrates that, for odd primes, Tate-Poitou duality can be enhanced to an Anderson duality between the homotopy fiber and the K(1)-local K-theory of the integers. In this talk, I will present this connection and extend the result to the case where p=2.

Abstract: I will talk about joint work with Kirsten Wickelgren on defining a global and local degree in stable equivariant homotopy theory. We construct the degree of a proper $G$-map between smooth $G$-manifolds and show a local to global property holds. This allows one to use the degree to compute topological invariants, such as the equivariant Euler characteristic and Euler number. I will discuss the construction of the equivariant degree and local degree, and I will give an application to counting orbits of rational plane cubics through 8 general points invariant under a finite group action on $\mathbb{C}\mathrm{P}^2$. This gives the first equivariantly enriched rational curve count, valued in the representation ring and Burnside ring. This equivariantly enriched count also recovers a Welchinger invariant in the case when $\mathbb{Z}/2$ acts on $\mathbb{C}\mathrm{P}^2$ by conjugation.

Abstract: I will talk about joint work with Corey Bregman and Jan Steinebrunner, in which we study the moduli space B Diff(M), for M a compact, connected, reducible 3-manifold. We prove that when M is orientable and has non-empty boundary, B Diff(M rel ∂M) has the homotopy type of a finite CW-complex. This was conjectured by Kontsevich and previously proved in the case where M is irreducible by Hatcher and McCullough.

Abstract: We develop a theory of schemes which come equipped with genuine involutions and develop invariants analogous to Picard and Brauer invariants for classical schemes. A key component of these invariants are Poincare structures on compact module categories. This is joint work with Noah Riggenbach and Lucy Yang.

Abstract: Classically, the Algebraic $K$-theory of spaces ($A$-theory) is used to study manifold topology from a homotopical perspective. In the equivariant setting, Malkiewich and Merling constructed a genuine $G$-spectrum $A_G(X)$ together with an assembly map $\Sigma^{\infty}_{G}X \to A_G(X)$ whose cofiber deloops to the equivariant "stable $h$-cobordism space" for a smooth $G$-manifold $M$. Non-equivariantly, Waldhausen’s original vision for $A$-theory was an interpretation that initiated work in "brave new algebra" that happens on the level of spectra. Moreover, he gave an interpretation of $A$-theory analogous to the theory of rings where we take the $K$-theory of $(\mathbb{S}[ \Omega X])$, thinking of this as a "spherical group ring" in analogy with $\mathbb{Z}[\pi_1 X]$ . A natural question is whether or not there is a similar story for $A_G(X)$, and we propose a model that gives a positive answer to this question. As an application, we construct an equivariant trace map to a version of equivariant topological Hochschild Homology possessing the correct properties in analogy with the identification of $THH(\mathbb {S}[\Omega X])$ as the free loop space of $X$.

Abstract: Motivic homotopy theory is the incarnation of classical homotopy theory in the world of algebraic geometry, and Betti realization is a tool that allows us to translate between motivic and classical homotopy theories. This tool is immensely useful for studying "transcendental" invariants of schemes, and was used by Voevodsky in his proof of the Bloch-Kato conjecture to compute the motivic Steenrod algebra. Just as there are equivariant versions of homotopy categories, in the past 10 years, there have been advances in motivic homotopy theory allowing us to consider equivariant versions of motivic homotopy categories. In this talk I will outline a way to produce equivariant and stacky versions of Betti realization allowing us to pass from equivariant motivic homotopy categories to classical equivariant homotopy categories. On the way, we will consider complex-analytic versions of motivic homotopy, and find models for equivariant homotopy categories using categories of homotopy-invariant sheaves.

Abstract: 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 associated to proper, locally constractible maps is then obtained by applying topological Hochschild homology to the categorified transfers.
If time permits, we will also explain how similar methods can be used to give a generalization of the topological Dwyer-Weiss-Williams index theorem, and how to construct analogous functorial transfers in any six functor formalism.
This is a joint work with Maxime Ramzi and Sebastian Wolf.