# EPFL Topology Seminar 2018/19

Tuesdays at 10:15

MA 12

### Program

Date Title Speaker
25.09.2018

in CM 012

The loop homology algebra of discrete torsion Yasuhiko Asao

University of Tokyo

09.10.2018 Topological Hochschild Homology and Fixed Point Invariants Jonathan Campbell

Vanderbilt University

16.10.2018 K-Theory and Polytopes Jonathan Campbell

Vanderbilt University

23.10.2018 Toward the equivariant stable parametrized h-cobordism theorem Mona Merling

University of Pennsylvania

30.10.2018 Tools for understanding topological coHochschild homology Anna Marie Bohmann

Vanderbilt University

06.11.2018 Thom spectra and Calabi-Yau algebras Inbar Klang

EPFL

13.11.2018
Periodic orbits and topological restriction homology
Cary Malkiewich

Binghamton University

20.11.2018 Hochschild homology and the de Rham complex, revisited Arpon Raksit

Stanford University

27.11.2018 Power operations on homology with twisted coefficients Calista Bernard

Stanford University

18.12.2018 Prederivators as a model of (∞,1)-categories Martina Rovelli

Johns Hopkins University

08.01.2019

in MA 12

Cohomology of braids, graph complexes, and configuration space integrals Ismar Volic

Wellesley

Universitat Autònoma de Barcelona

26.02.2019 An Axiomatic Approach to Algebraic Topology Nima Rasekh

MPIM Bonn

05.03.2019 Partial categories and directed path spaces — a proposed construction of orbit categories for p-local finite groups Sune Precht Reeh

Universitat Autònoma de Barcelona

12.03.2019 Homotopical Galois extensions of E_\infty algebras Magdalena Kedziorek

Universiteit Utrecht

26.03.2019 A new approach to the Dold-Thom theorem Lauren Bandklayder

MPIM Bonn

02.04.2019 Field theories in synthetic differential geometry Melvin Vaupel

ETH Zurich

07.05.2019 Markus Hausmann

University of Copenhagen

21.05.2019 Brenda Johnson

Union College

28.05.2019 Mingcong Zeng

Universiteit Utrecht

04.06.2019 Christian Ausoni

Université Paris 13

11.06.2019 Kate Ponto

University of Kentucky

(See also the program of the topology seminar in 2011/12,  2010/11,  2009/102008/09,  2007/08, 2006/07, and 2005/06.)

### Abstracts

Yasuhiko Asao – The loop homology algebra of discrete torsion

Let $M$ be a closed oriented manifold with a finite group action by $G$.
We denote its Borel construction by $M_{G}$. As an extension of string
topology due to Chas-Sullivan, Lupercio-Uribe-Xicot$\’{e}$ncatl
constructed a graded commutative associative product (loop product) on $H_{*}(LM_{G})$, which plays a significant role in the “orbifold string
topology” . They also showed that the constructed loop product is an
orbifold invariant. In this talk, we describe the orbifold loop product
by determining its “twisting” out of  the ordinary loop product in term
of the group cohomology of $G$, when the action is homotopically trivial.
Through this description, the orbifold loop homology algebra can be
seen as R. Kauffmann’s “algebra of discrete torsion”, which is a group
quotient object of Frobenius algebra. As a cororally, we see that the
orbifold loop product is a non-trivial orbifold invariant.

Jonathan Campbell –  Topological Hochschild Homology and Fixed Point Invariants

Fixed point theorists have an array of invariants for determining if a self-map is homotopic to one without fixed points – most are elaborations of the classical Lefschetz invariant. In this talk, I’ll discuss these invariants and show that they arise very naturally from topological Hochschild homology. Furthermore, the invariants are in the image on components of the cyclotomic trace from K-theory to THH. This suggests higher K-theory and THH as homes for more refined fixed-point data. In order to make the talk accessible I’ll define the objects involved – a knowledge of the stable homotopy category is the only prerequisite. This is joint work with Kate Ponto.

Jonathan Campbell – K-Theory and Polytopes

In this talk I’ll describe joint work in progress with Inna Zakharevich to understand the relationship between algebraic K-theory and the scissors congruence problem. I’ll define both algebraic K-theory and the scissors congruence problem, and then describe various ways of using the former to attack the latter. And possible ways that the latter can shed light on the former.

Mona Merling –  Toward the equivariant stable parametrized h-cobordism theorem

Waldhausen’s introduction of A-theory of spaces revolutionized the early study of pseudo-isotopy theory. Waldhausen proved that the A-theory of a manifold splits as its suspension spectrum and a factor Wh(M) whose first delooping is the space of stable h-cobordisms, and its second delooping is the space of stable pseudo-isotopies. I will describe a joint project with C. Malkiewich aimed at telling the equivariant story if one starts with a manifold M with group action by a finite group G.

Anna Marie Bohmann –  Tools for understanding topological coHochschild homology

Hochschild homology is a classical invariant of algebras.  A “topological” version, called THH, has important connections to algebraic K-theory, Waldhausen’s A-theory, and free loop spaces.  For coalgebras, there is a dual invariant called “coHochschild homology” and Hess and Shipley have recently defined a topological version called “coTHH,” which also has connections to K-theory, A-theory and free loops spaces.  In this talk, I’ll talk about coTHH (and THH) are defined and then discuss work with Gerhardt, Hogenhaven, Shipley and Ziegenhagen in which we develop some computational tools for approaching coTHH.

Inbar Klang –  Thom spectra and Calabi-Yau algebras

I’ll briefly introduce topological field theories, along with some examples of 2-dimensional ones. I’ll discuss how 2-dimensional TFTs are related to Calabi-Yau algebras, which are algebras that satisfy a certain dualizability condition. Then I will talk about joint work with Ralph Cohen, in which we defined and studied a spectrum-level generalization of the Calabi-Yau condition, as well as its relation to field theories and to symplectic geometry.

Cary Malkiewich – Periodic orbits and topological restriction homology

I will talk about a project to import trace methods, usually reserved for algebraic K-theory computations, into the study of periodic orbits of continuous dynamical systems (and vice-versa). Our main result so far is that a certain fixed-point invariant built using equivariant spectra can be “unwound” into a more classical invariant that detects periodic orbits. As a simple consequence, periodic-point problems (i.e. finding a homotopy of a continuous map that removes its n-periodic orbits) can be reduced to equivariant fixed-point problems. This answers a conjecture of Klein and Williams, and allows us to interpret their invariant as a class in topological restriction homology (TR), coinciding with a class defined earlier in the thesis of Iwashita and separately by Luck. This is joint work with Kate Ponto.

Arpon Raksit –  Hochschild homology and the de Rham complex, revisited

I will describe a conceptual perspective on the story relating Hochschild homology and the algebraic de Rham complex in the setting of commutative rings. A bonus of this perspective is that it supplies a variant of the story in the setting of E-infinity algebras over the integers. The rough idea is as follows: by considering certain types of structure in higher algebra, we may give universal properties to the derived de Rham complex and the Hochschild-Kostant-Rosenberg filtration on Hochschild homology (and their analogues for E-infinity algebras); these objects are then related directly by these universal properties.

Calista Bernard –  Power operations on homology with twisted coefficients

The structure of an E_n algebra on a space X gives rise to power operations on the homology of X generated by the so-called Dyer-Lashof-Kudo-Araki operations and a Browder bracket. This proves very useful for calculations; however, these operations are only defined for ordinary mod p homology, and in many instances, it is desirable to have an analogue of these operations for homology with twisted coefficients. In this talk I will give an overview of these operations for untwisted mod p homology, followed by a description of ongoing work constructing similar operations on the homology of E_2 algebras with certain twisted coefficient systems.

Martina Rovelli –  Prederivators as a model of (∞,1)-categories

By theorems of Carlson and Renaudin, the theory of (∞,1)-categories embeds in that of prederivators. We present two approaches to address the inverse problem: understanding which prederivators model (∞,1)-categories. First, we put a model structure on the category of prederivators that is Quillen equivalent to the Joyal model structure for quasi-categories. Second, we give an intrinsic description of which prederivators arise on the nose as prederivators associated to quasi-categories. This is joint work with D. Fuentes-Keuthan and M. Kędziorek.

Ismar Volic –  Cohomology of braids, graph complexes, and configuration space integrals

In this talk, I will explain how three integration techniques for producing cohomology — Chen integrals for loop spaces, Bott-Taubes integrals for knots and links, and Kontsevich integrals for configuration spaces — come together in the computation of the cohomology of spaces of braids. The relationship between various integrals is encoded by certain graph complexes. I will also talk about the generalizations to other spaces of maps into configuration spaces (of which braids are an example), and this will lead to connections to spaces of link maps and, from there, to manifold caclulus of functors. This is joint work with Rafal Komendarczyk and Robin Koytcheff.

I’ll present a new model for ∞-operads, namely as analytic monads. In the ∞-world (unlike what happens in the classical case), analytic functors are polynomial, and therefore the theory can be developed within the setting of polynomial functors. I’ll talk about some of the features of this theory, and explain a nerve theorem, which implies that the ∞-category of analytic monads is equivalent to the ∞-category of dendroidal Segal spaces of Cisinski and Moerdijk, one of the known equivalent models for ∞-operads.  This is joint work with David Gepner and Rune Haugseng.

Nima Rasekh –  An Axiomatic Approach to Algebraic Topology

An elementary higher topos is a higher category that is defined using only elementary conditions, yet behaves similar to the category of spaces. The goal of this talk is to illustrate this connection by proving classical results from algebraic topology in this abstract setting. Concretely, we will use the fact that it satisfies descent, which a kind of a local-to-global condition, to construct natural number objects. This allows us to use inductive arguments. Using induction, we will then construct truncations and show that we can also prove the Blakers-Massey theorem.

Sune Precht Reeh –  Partial categories and directed path spaces — a proposed construction of orbit categories for p-local finite groups

Given a finite group G the orbit category of G consists of all transitive G-sets and equivariant maps between them. Aside from algebra the orbit category has also proved very useful in topology for describing G-equivariant homotopy theory.
For a saturated fusion system/p-local finite group F with Sylow subgroup S the existing constructions of an orbit category for F only makes sense for subgroups of S that are “sufficiently large”. In this talk I will propose a construction of an orbit category for F that works for all subgroups of S, but the result will be a “partial” category (to be defined during the talk) where composition of morphisms is only partially defined. The construction builds upon the theorem of Andy Chermak that a saturated fusion system is always realized by a (suitably unique) partial group.
Every partial category gives rise to a actual category enriched in simplicial sets via a very explicit procedure using a sort of directed path spaces. We shall see that the proposed orbit category restricts to a classical category when considering “sufficiently large” subgroups of S and that we recover the old definition of an orbit category for F.
This joint project with Rémi Molinier is very much a work in progress, and as such the talk will contain many more conjectures than theorems.

Magdalena Kedziorek –  Homotopical Galois extensions of E_\infty algebras

Rognes generalised the classical notion of Galois extensions from algebra to homotopy theory of ring spectra. In recent work with Beaudry, Hess, Merling and Stojanoska we discussed a general formal framework in which the notion of (homotopical) Galois extensions can be studied and we applied that work to the motivic homotopy theory.
In this talk I will recall the general framework for homotopical Galois extensions and present an ongoing work with Hess on homotopical Galois theory for E_\infty – algebras.

Lauren Bandklayder –  A new approach to the Dold-Thom theorem

The Dold-Thom theorem is a classical result in algebraic topology giving isomorphisms between the homology groups of a space and the homotopy groups of its infinite symmetric product. The goal of this talk is to outline a new proof of this theorem, which is direct and geometric in nature. The heart of this proof is a hypercover argument which identifies the infinite symmetric product as an instance of factorization homology.

Melvin Vaupel – Field theories in synthetic differential geometry

First I will review the construction of the Cahiers topos and explain why it is a well adapted model for synthetic differential geometry. The Cahiers topos is a sheaf topos on the category of generalized smooth spaces with a certain Grothendieck topology.
Next I will explain how the Cahiers topos can be used to formulate scalar field theories in synthetic differential geometry.
To handle the gauge transformations of Yang-Mills theory it is more natural to work with (pre-)sheaves of generalized smooth spaces that take values in Grpd rather than in Set. We are interested in the stacks of this presheaf category. Hollander identified them as the fibrant objects with respect to a certain localized model structure. I will sketch how these stacks of generalized smooth spaces might be used to formulate Yang-Mills theory in synthetic differential geometry.

Markus Hausmann –  TBA

TBA

Brenda Johnson –  TBA

TBA

Mingcong Zeng –  TBA

TBA

Christian Ausoni –  TBA

TBA

Kate Ponto –  TBA

TBA