# Algebraic Topology Seminar

The current seminar is here.

###### Term 3

The talks will take place on Tuesday on MS teams at 4pm. You can join the team with the code **hud33su**, or email me at emanuele.dotto@warwick.ac.uk.

Date | Speaker | Affiliation | Title | Notes |
---|---|---|---|---|

Apr 21 | ||||

Apr 28 | Jan Steinebrunner | Oxford |
The classifying space of the one-dimensional bordism category and a cobordism model for TC of simply connected spaces |
Watch Video |

May 5 | Markus Hausmann | Bonn | Global group laws and the equivariant Quillen theorem | Watch Video |

May 12 | Niall Taggart | Belfast | Unitary functor calculus with Reality | Watch Video |

May 19 | Roy Joshua | Ohio State | The Motivic and Etale Becker-Gottlieb transfer and Applications | |

May 27 | Achim Krause | Muenster | The Picard group in equivariant homotopy theory via Stable Module categories | Watch Video |

Jun 2 | John Jones | Warwick | Derivations of Lie algebras and homotopy equivalences | Watch Video |

Jun 9 | Frank Neumann | Leicester | On the cohomology of classifying stacks for some algebraic groups | Watch Video |

Jun 16 | Gijs Heuts | Utrecht | Koszul duality of algebras for operads | Watch Video |

Jun 23 | George Raptis | Regensburg | Higher homotopy categories and K-theory |

##### Abstracts

###### Jan Steinebrunner: The classifying space of the one-dimensional bordism category and a cobordism model for TC of simply connected spaces

The study of bordism categories and their classifying spaces has proven extremely useful, for example in the study of moduli spaces of manifolds. Usually, this approach uses a topologically enriched bordism category Cob_{d}, but in this talk I will be looking at the much simpler homotopy category h(Cob_{d}) where diffeomorphic bordisms are identified. I will begin by recalling both notions and how they differ.

Using a new fiber sequence for bordism categories I will compute the classifying space of h(Cob_{1}) in terms of the spectrum MTSO_{2}. This spectrum, also known as CP_{-1}^{∞}, is related to both Topological Cyclic Homology (TC) and the stable mapping class group Γ_{∞}. Both areas show up in the 1-dimensional bordism category. To see this, I will construct a `reduced' variant of Cob_{1} who’s classifying space (in the presence of a simply connected background space X) is essentially the TC of the spherical group ring of X. If time permits, I will also sketch how h(Cob_{1}) is related to Connes' cyclic category Λ and use this describe cocycles on h(Cob_{1}) representing the Miller-Morita-Mumford classes κ_{i}.

###### Markus Hausmann: Global group laws and the equivariant Quillen theorem

Quillen's theorem that the complex bordism ring carries the universal formal group law is a fundamental result in stable homotopy theory. In this talk I will discuss equivariant versions of this theorem, both over a fixed abelian compact Lie group and in a global equivariant setting.

###### Niall Taggart: Unitary functor calculus with Reality

Unitary functor calculus is a method of studying functors from the category of complex vector spaces to spaces, a variation of the orthogonal calculus developed by Weiss in the 1990s. In this talk, I will describe the construction a calculus of functors in this spirit, which is designed to study ``unitary functors with reality” such as the functor which sends a vector space to the Real classifying space of its unitary group. This calculus is the first step in understanding a “genuine” equivariant version of orthogonal calculus. The calculus produces a Taylor tower, the $n$-th layer of which is classified by a spectrum with an action of $C_2 \ltimes U(n)$. I will explain this classification both on the point-set and model categorical level.

###### Roy Joshua: The Motivic and Etale Becker-Gottlieb transfer and Applications

In this talk, which is based on joint work with Gunnar Carlsson and also Pablo Pelaez, we discuss a theory of Becker-Gottlieb transfer based on Spanier-Whitehead duality that holds in both the motivic and étale settings for smooth quasi-projective varieties in as broad a

context as possible: for example, for varieties over non-separably closed fields in all characteristics, and also for both the étale and motivic settings.

In view of the fact that the most promising applications of the traditional Becker-Gottlieb transfer has been to torsors and Borel-style equivariant cohomology theories, we focus our applications to motivic cohomology theories for torsors as well as Borel-style equivariant motivic cohomology theories, both defined with respect to motivic spectra.

###### Achim Krause: The Picard group in equivariant homotopy theory via Stable Module categories

The Picard group of genuine G-spectra consists of those compact objects all of whose geometric fixed points are spheres. Work by tom Dieck-Petrie almost classifies those objects in terms of the dimensions of their geometric fixed points, but it is not fully known what the realizable dimensions are. In this talk, we present a new approach to compute these Picard groups, based on a different perspective on isotropy separation that interacts well with compact objects.

###### John Jones: Derivations of Lie algebras and homotopy equivalences

Let X be a finite CW complex and let G(X) be the space of self maps of X that are homotopic to the identity — G(X) is topologized as a sub-space of the space of all self maps of X with its “usual” topology. This space is not a topological group but it is “group-like” in a precise sense. Therefore it’s homotopy groups become a graded Lie algebra, where the bracket is the Samelson product. The aim of this talk is to study this Lie algebra in the case where X is the configuration space F(k, n) of k distinct ordered points in Euclidean n-space. The main point is to show how this Lie algebra is related to Ihara’s stable derivation algebra (also known as the Grothendieck - Teichmuller Lie algebra).

###### Frank Neumann: On the cohomology of classifying stacks for some algebraic groups

We compute the number of rational points of classifying stacks of some Chevalley groups using Behrend’s Lefschetz-Grothendieck trace formula for l-adic cohomology of algebraic stacks and derive some associated zeta function. On the way we show how classical results for the cohomology of compact Lie groups can be resurrected in this more algebraic setting which might be of independent interest (joint work with Scott Balchin (Warwick)).

###### Gijs Heuts: Koszul duality of algebras for operads

Ginzburg-Kapranov and Getzler-Jones exhibited a duality between algebras for an operad O and coalgebras (with divided powers) for a "Koszul dual" cooperad BO, taking the form of an adjoint pair of functors between these categories. Instances of this duality include that between Lie algebras and cocommutative coalgebras, as in Quillen's work on rational homotopy theory, and bar-cobar duality for associative (co)algebras, as in the work of Moore. I will review this formalism and discuss the following basic question: on what subcategories of O-algebras and BO-coalgebras does this duality adjunction restrict to an equivalence? I will discuss an answer to this question and explain the relation to a conjecture of Francis and Gaitsgory.

###### George Raptis: Higher homotopy categories and K-theory

I will discuss the construction and the properties of the higher homotopy categories associated to an infinity-category. These objects may be regarded as a sequence of refinements for the comparison between homotopy commutativity and homotopy coherence. Even though the idea of the higher homotopy category is not new, the study of these objects seems to have received less attention than the classical homotopy category. I will introduce K-theory for these objects and present some results on the comparison with Waldhausen K-theory. If time permits, I will also discuss generalizations of Grothendieck derivators and derivator K-theory to (n,1)-categories, and present some analogous results about the comparison with Waldhausen K-theory.

###### Term 2

The talks take place on Tuesday at 4pm, in room B3.01 (in the Zeeman building), unless noted otherwise.

Date | Speaker | Affiliation | Title | Notes |

Jan 7 | Dave Barnes | Belfast | Mackey functors and equivariant sheaves for profinite groups | at 2:30 in LIB 2 |

Jan 14 | Manuel Krannich | Cambridge | A homological approach to pseudo-isotopy theory | |

Jan 21 | No Seminar | |||

Jan 28 | Scott Balchin | Warwick | The module adelic model for a tensor-triangulated category | |

Feb 4 | No Seminar | |||

Feb 11 | Matteo Barucco | Warwick | An algebraic model for rational equivariant elliptic cohomology | |

Feb 18 | Irakli Patchkoria | Aberdeen | Equivariant dimensions for groups, LS category, and cohomology of orbit categories | |

Feb 25 | Tom Bachmann | MIT | Eta-periodic motivic stable homotopy theory | |

Mar 3 | Igor Sikora | Warwick | C_2x\Sigma_n-equivariant Bredon homology of equivariant little discs operads | |

Mar 10 | Mauricio Bustamante | Cambridge | The Nielsen realization problem for exotic negatively curved manifolds |

##### Abstracts

###### Dave Barnes:

Mackey functors occur naturally in representation theory and algebraic topology (via equivariant cohomology theories). For finite $G$, these have been long-studied and are the category has a very rich structure. Examples include representation rings, Burnside rings and the homotopy groups of G-spectra. If we take rational coefficients, the category of $G$-Mackey functors has a much simpler description in terms of modules over group rings of Weyl groups.

This talk focuses on the generalisations of these results to the case of profinite groups. A profinite group G is a compact Hausdorff group that is totally disconnected. Any such group is the (filtered) inverse limit of a diagram of finite groups. This causes several complications, primarily that the rational Burnside ring (the colimit of the finite groups in the system for G) is no longer a finite product of copies of the rationals. However, a classification of rational G-Mackey functors is still possible, using the technology of equivariant sheaves to give the description.

###### Manuel Krannich:

There is an intimate connection between algebraic K-theory and the space of pseudo-isotopies P(M) of a compact d-manifold M (that is, diffeomorphisms of a cylinder M x I that are the identity on M x 0 and ∂M x I). Classically, the pseudo-isotopy space P(M) is studied in two steps: there is a stabilisation map P(M)-> P(M x I) which is approximately d/3-connected by a result of Igusa, and the colimit has a description in terms of Waldhausen's algebraic K-theory of spaces due to Waldhausen--Jahren--Rognes' stable parametrised h-cobordism theorem.

In this talk, I will focus on the case of an even-dimensional disc and explain a new method to relate its space of pseudo-isotopies to the algebraic K-theory of the integers, independent of the usual route. When combined with work of Goodwillie and of Randal-Williams, one consequence of this approach is that the stabilisation map P(M) ->P(M x I) for 2-connected d-manifolds M is rationally about d-connected, which is essentially optimal.

###### Scott Balchin:

(j/w J.P.C. Greenlees) Tensor-triangulated categories naturally occur in various settings, for example, as the derived category of a commutative ring, the category of (possibly equivariant, possibly rational) cohomology theories, or the stable module category of a Frobenius ring. To any such category, one can assign the so-called Balmer spectrum, which is a categorification of the Zariski spectrum of a ring. I will report on joint work with J.P.C. Greenlees which provides a machinery to build up the unit object of the category from smaller building blocks governed by data of the Balmer spectrum. This fracturing is then used to build a model for the original category, based on modules over each building block. Not only does this result give an insight into the structure of the category, but it usually provides a convenient setting to do calculations in.

###### Matteo Barucco:

Important cohomology theories are associated to one dimensional Group Schemes. Ordinary cohomology is associated to the additive group, while K-Theory is associated to the multiplicative group. Following this correspondence it's therefore natural to consider cohomology theories associated to elliptic curves and try to construct equivariant versions of them. One of the first applications of the algebraic models for rational S^1-spectra introduced by Greenlees was the construction of an algebraic model for rational S^1-equivariant elliptic cohomology. After an historical overview of the problem of defining a satisfactory notion of equivariant elliptic cohomology, we will construct this model discussing which aspects of the theory are reflected in the geometry of the elliptic curve.

###### Irakli Patchkoria:

The classical Eleinberg-Ganea theorem compares the cohomological dimension and geometric dimension of a discrete group G with the Lusternik-Schnirelmann category of the classifying space BG. On the other hand the Stallings-Swan theorem says that a group has cohomological dimension 1 if and only if it is free. In this talk we discuss equivariant generalisations of these results for groups with operators. Along the way we prove an algebraic result which says that the cohomological dimension of a finite group with respect to a family consisting of proper subgroups (which by definition is the cohomological dimension of the correpsonding orbit category) is bigger than 1. This verifies a special case of the conjecture which says that cohomological dimension of a group with respect to any family is 1 if and only if its geometric dimension is 1 with respect to this family. The talk will introduce basic concepts at the beginning and should be accessible to a general audience. This is all joint work with Mark Grant and Ehud Meir.

###### Tom Bachmann:

(Report on work in progress joint with Mike Hopkins) The motivic spectrum ko[1/3] (related to hermitian K-theory) carries an Adams operation psi^3, and the difference psi^3-1 factors through the "very 4-effective cover" Sigma^{4,2} ksp[1/3] of ko[1/3]. Form the fiber sequence

jo-->ko_{(2)} --> Sigma^{4,2} ksp_{(2)}.

Then jo is a "motivic image of J" spectrum, considered previously by Quigley-Culver and Ormsby-Röndigs. Our main result is that the unit map S_{(2)} ->jo is an eta-periodic equivalence. As applications, we compute the eta-periodic motivic stable stems over fields of characteristic not two (extending results of Andrews-Miller, Guillou-Isaksen and Ormsby-Röndigs) as well as the eta-periodic algebraic special linear and symplectic cobordism groups (over the same fields).

###### Igor Sikora:

Loop spaces occur naturally in many places in topology. Adjointness of nth suspension and n-fold loops functor provide a map from a space X to n-fold loop space of its n-th suspension. It is a natural question to ask about homology of this space in terms of homology of X - which was answered by Araki, Kudo, Dyer, Lashof and most notably May and Cohen. Homology of loop spaces with coefficients in finite field have also rich algebraic structure, similar to action of Steenrod algebra on cohomology. This structure is encoded in the action of a little discs operad.

In the talk, I will aim to provide a description of part of equivariant counterpart of the story above. After starting with definitions of Bredon homology and equivariant operads, I will talk about methods of computing this homology of operads with respect to actions of C_2 and \Sigma_n.

###### Mauricio Bustamante:

Let N be a smooth manifold that is homeomorphic but not diffeomorphic to a closed hyperbolic manifold M. To what extent does N admit as much symmetry as M? In this talk, I will show that it's possible to find N with maximal symmetry, i.e. Isom(M) acts on N by isometries with respect to some negatively curved metric on N. For these examples, Isom(M) can be made arbitrarily large. On the other hand, one can also find N with little symmetry, i.e. no subgroup of Isom(M) of “small” index acts by diffeomorphisms of N. The construction of these examples incorporates a variety of techniques including smoothing theory and the Belolipetsky-Lubotzky method for constructing hyperbolic manifolds with a prescribed isometry group. This is joint work with Bena Tshishiku.

###### Term 1

Date | Speaker | Affiliation | Title | Notes |

Oct 22 | Dave Benson | Aberdeen | Hochschild cohomology of finite groups | |

Oct 29 | Lukas Brantner | Oxford | Deformation theory and Lie algebras in characteristic p | |

Nov 5 | Luca Pol | Sheffield | Cofree G-spectra and completions | |

Nov 12 | Jordan Williamson | Sheffield | Flatness and change of groups functors | |

Nov 19 | Emanuele Dotto | Warwick | The Witt vectors with coefficients | |

Nov 26 | Richard Hepworth | Aberdeen | Stability for Coxeter, Artin and Iwahori-Hecke | |

Dec 3 | Clark Barwick | Edinburgh | Compactifying arithmetic schemes |
2 parts talk: 2-3 in MS.05 4-5 in |

##### Abstracts

###### Dave Benson:

I shall introduce Hochschild homology and cohomology and explain what they look like for a group algebra. Hochschild cohomology admits a Gerstenhaber bracket, and in the case of the group algebra of a finite group, there is a Batalin-Vilkovisky (BV) operator with the property that if we know the product structure and the BV operator we can compute the Gerstenhaber bracket. I shall describe how to compute this BV operator, and give some consequences for the structure of the Lie algebra of derivations of the group algebra. This is joint work with Kessar and Linckelmann.

###### Lukas Brantner:

The infinitesimal structure of any moduli space in characteristic zero is controlled by a differential graded Lie algebra. This far-reaching generalisation of Kodaira-Spencer theory was established by Lurie and Pridham, based on previous work of Deligne, Drinfeld, Feigin, Hinich, and others. In this talk, I will explain how to generalise the above statement to finite and mixed characteristic. This is joint work with Akhil Mathew.

###### Luca Pol:

###### Jordan Williamson:

The behaviour of the forgetful functor from commutative algebra spectra to module spectra is subtle. In the usual stable model structure, this functor does not preserve cofibrant objects. However, Shipley constructs a flat model structure in which this functor does preserve cofibrant objects. I will explain how this turns out to be a vital ingredient in understanding the relation between the induction-restriction-coinduction adjunction between categories of rational equivariant cohomology theories, and functors in algebra.

###### Emanuele Dotto:

We will introduce the Witt vectors of a ring with coefficients in a bimodule and use them to calculate the components of the Hill-Hopkins-Ravenel norm for cyclic p-groups. This algebraic construction generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors over perfect fields. We will discuss applications to the characteristic polynomial over non-commutative rings and to the Dieudonné determinant. This is joint work with Achim Krause, Thomas Nikolaus and Irakli Patchkoria.

###### Richard Hepworth:

Homological stability is a topological property that is satisfied by many families of groups, including the symmetric groups, braid groups, general linear groups, mapping class groups and more; it has been studied since the 1950's, with a lot of current activity and new techniques. In this talk I will explain a set of homological stability results from the past few years, on Coxeter groups, Artin groups, and Iwahori-Hecke algebras (some due to myself and others due to Rachael Boyd). I won't assume any knowledge of these things in advance, and I will try to introduce and motivate it all gently!

###### Clark Barwick

Half a century ago, Barry Mazur launched the industry known today as ‘arithmetic topology’ by observing that the ‘homotopy theory’ of a number ring O_F closely resembles that of a 3-manifold. Indeed, class field theory implies a kind of 3-dimensional Poincaré Duality for the étale cohomology of O_F. Unfortunately, this analogy has always suffered from deficiencies with real places and with 2-local coefficients. We propose a different, homotopical compactification, which enjoys a 3-dimensional duality theorem with no fudging around real places and the prime 2. The compactification is closely linked to the geometrisation of Galois and Weil groups, which we will also discuss.