EPFL Topology Seminar 2018/19

Tuesdays at 10:15

MA 12

Program

(See also the program of the topology seminar in 2011/12,  2010/11,  2009/10, 2008/09,  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-Xicoténcatl
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.

Joachim Kock –  Infinity-operads as polynomial monads

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

Magdalena Kedziorek –  Homotopical Galois extensions of 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 –  Chromatic Smith theory

Let X be a finite complex equipped with an action of a finite group G. Classical Smith theory studies the relationship between the mod p homology of X and the mod p homology of the fixed point space X^G. In this talk I will describe a variant of this theory where mod p homology is replaced by other generalized homology theories. In particular I will discuss a relationship between the chromatic types of X and  X^G, and how this is related to chromatic blueshift of Tate constructions, partition complexes and the Balmer spectrum of finite G-spectra.

Brenda Johnson –  An Introduction to Functor Calculus

There are many kinds of functor calculi that have been developed to study functors in a wide range of algebraic and topological contexts. In this talk I will focus on three examples of functor calculi – the calculus of homotopy functors, manifold calculus, and abelian functor calculus – and discuss how they are defined, the properties they have in common, and how they capture certain features of polynomial functions and Taylor series from undergraduate calculus. As the title suggests, this will be an introductory talk setting up subsequent talks on work arising from these calculi.

Mingcong Zeng –  Real cobordism, its norm and the dual Steenrod algebra

The real cobordism spectrum MU_R and its norms play a central role in the proof of the non-existence of classes of Kervaire invariant one by Hill, Hopkins and Ravenel. However, these spectra are still very mysterious and their equivariant homotopy groups are difficult to compute.

In this talk I will focus on the norm of real cobordism into C_4, and draw a connection between it and the dual Steenrod algebra spectrum HF_2 \smash HF_2 with C_2-action by conjugation.  Then I will discuss how computations on both sides help each other.

This is joint work with Lennart Meier.

Christian Ausoni –  On the Hochschild homology of Johnson-Wilson spectra

Let E(n) denote the n-th Johnson-Wilson spectrum at an odd prime p.
The spectrum E(1) coincides with the Adams summand of p-local topological K-theory.
McClure and Staffeldt offered an intriguing computation
of THH(E(1)), showing that it splits as a wedge sum of E(1) and a rationalized suspension of E(1).

In joint work with Birgit Richter, we study the Morava K-theories of THH(E(n)),
with an aim at investigating if McClure-Staffeldt’s splitting in lower chromatic
pieces generalizes.  Under the assumption that E(2) is commutative, we show that
THH(E(2)) splits as a wedge sum of E(2) and its lower chromatic localizations.

Kate Ponto –  Defining additive invariants

I’ll talk about first steps in an approach to defining additive invariants that captures familiar invariants including the character of a representation and the Reidemister trace of an endomorphism of a compact manifold. One advantage of this approach is that many of the most useful properties of these invariants are consequences of a very formal structure. These include induction and restriction formulas for characters and the compatibility of fixed point invariants with maps of subcomplexes and fibrations.

Lyne Moser –  An adventure toward (∞,2)-limits

The theory of (∞,1)-categories has been extensively studied until now. Several models of (∞,1)-categories have been developed, and (∞,1)-limits have been constructed in each model, as terminal objects in the (∞,1)-category of cones. I will focus on Rezk’s model of (∞,1)-categories, called complete Segal spaces, and explain the (∞,1)-limit construction in this model. The aim is to generalize this construction to (∞,2)-categories.

However, some difficulties arise when we go from the case n=1 to the case n=2. In a 2-categorical setting, one can not define a 2-limit as a terminal object in a 2-category of cones. Instead, one needs to consider additional structure on a 2-category, and consider double categories. Then, it is true that a 2-limit is a terminal object in the double category of cones, where 2-categories are regarded as horizontal double categories.

To adapt this to the ∞-setting, I will explain how to define a notion of double (∞,1)-categories, as simplicial objects in complete Segal spaces, and how to regard an (∞,2)-category as a horizontal double (∞,1)-category. This provides a framework for defining (∞,2)-limits.

Anibal Medina –  A finitely presented $E_\infty$-prop

Commutativity up to coherent homotopies is a central concept in topology, which after Boardman-Vogt and May is described using the notions of $E_\infty$-prop and $E_\infty$-operad. Many models have been constructed for the $E_\infty$-operad in several categories with a notion of homotopy theory, each having its range of applicability, advantages, and drawbacks. I will describe work motivated by the search of a model as small as possible in terms of generators and relations. No finite presentation for a model of the $E_\infty$-operad can exist, but passing to the more general setting of $E_\infty$-props allows for one such presentation. We will discuss this construction in the category of chain complexes and use it to endow the chains of simplicial sets with an $E_\infty$-structure generated by the AW diagonal, the augmentation map, and a dg version of the join map. We will also describe a lift of this construction to the category of CW-complexes and use it to prove a conjecture of R. Kaufmann.

Aloïs Rosset –  Blakers-Massey Connectivity Theorem from the perspective of homotopy type theory

The Blakers-Massey Connectivity Theorem is an important result of homotopy theory. It expresses the high connectivity of a specific map going into a pullback.
Homotopy type theory is a field which combines type theory and homotopy theory. It proposes a refreshing point of view on the way to construct and study mathematics.
The aim will be to present the specificities of the theory and to see how the usual notions and tools of homotopy theory can be adapted and translated in it, in order to look at the Blakers-Massey theorem.

Eric Finster –  A Topos Theoretic View of Goodwillie’s Calculus of Functors

One of the motivating ideas for Grothendieck’s introduction of the notion of a topos was that the theory of topoi served as a natural generalization of the notion of topological space: every topological space gives rise to a topos (its topos of sheaves), even if not every topos arises from this construction.  Using this intuition as a guide, a subtopos F of a topos E may be regarded as corresponding to a subspace.  In the case of ordinary
topoi, all such subtopoi are obtained from the theory of Grothendieck topologies, which can be regarded as a combinatorial way of parameterizing all the finite limit preserving localizations of the original topos E.

The higher topoi of Rezk and Lurie extend this picture to the world of homotopy theory by allowing one to consider not merely sheaves of sets on a space, but rather sheaves of homotopy types. The theory of subtopoi of higher topoi, however, turns out to hold some surprises.  First of all, it is no longer true that all subtopoi may be described by Grothendieck topologies. Furthermore the subtopoi of a higher topos carry additional “geometric” information which makes them somewhat richer than the notion of bare subspace.  In particular, we may speak of the formal neighborhoods of a subtopos and well as its formal completion.  I will explain how to view Goodwillie’s calculus of homotopy functors as well as the orthogonal calculus
of Michael Weiss as examples of this construction.