Topology seminars: 2022-23

We define generalised higher Whitehead maps in polyhedral products. By investigating the interplay between the homotopy-theoretic properties of polyhedral products and the combinatorial properties of simplicial complexes, we describe new families of relations among these maps, while recovering and generalising known identities among Whitehead products.

This is joint work with George Simmons and Matthew Staniforth.

In this mostly expository talk. I'll explain some (different) recipes for defining concepts of "polynomial map" and "polynomial functor" in various settings. I'll explain what some of this has to do with algebraic K-theory, and mention several things I don't know.

Configuration spaces are, on the one hand, powerful invariants and, on the other, spaces with many computable properties. They have also been shown to provide concrete models for homotopy-theoretical constructions such as the free E_n-algebras and the infinite loop spaces associated to stable homotopy theory. These spaces have been generalised in (at least) two directions: the first allows for controlled interactions between the particles of the configuration (for instance allowing some collisions), and the other looks at configurations not of points, but of more general submanifolds. In this talk we will discuss these generalisations, and how they lead to powerful constructions such as factorization homology. We will also discuss in which cases these spaces still carry desirable computational properties seen in the classical configuration spaces, such as homological stability.

The overall project is to build an algebraic model for rational G-equivariant cohomology theories for all compact Lie groups G. When G is small or abelian this has been done. In general, the model is expected to take the form of a category of sheaves of modules over a sheaf of rings over the space of closed subgroups of G. The talk will focus on structural features of the expected model for general G such as those above, and feature recent joint work with Balchin and Barthel.

For a category E with finite limits and well-behaved countable coproducts, we construct a new Quillen model structure on the category of simplicial objects in E, which we call the effective model structure. The effective model structure generalises the Kan-Quillen model structure on simplicial sets; in particular, its fibrant objects can be viewed as infinity-groupods (i.e. Kan complexes) in E. After introducing the main definitions and outlining the key steps of the proof of the existence of the effective model structure, I will describe some of its peculiar properties and what they mean in terms of its associated infinity-category. This is based on joint work with Simon Henry, Christian Sattler and Karol Szumiło (https://doi.org/10.1017/fms.2022.13).

The study of the action of a finite p-group G on a finite G-CW complex X is one of the oldest topics in algebraic topology.  In the late 1930's, P. A. Smith proved that if X is mod p acyclic, then so is X^G, its subspace of fixed points. A related theorem of Ed Floyd from the early 1950's says that the dimension of the mod p homology of X will bound the dimension of the mod p homology of X^G. 

The study of thick tensored categories in the category of G-spectra has led to the problem of identifying "chromatic" variants of these theorem, with mod p homology replaced by the Morava K-theories (at the prime p).  An example of a new chromatic Floyd theorem is the following: if G is a cyclic p-group, then the dimension over K(n)* of K(n)*(X) will bound the dimension over K(n-1)* of K(n-1)*(X^G).

These chromatic fixed point theorems open the door for new applications.  For example, one can deduce that a C_2 action on the 5 dimensional Wu manifold will have fixed points that have the rational homology of a sphere.  In a different direction, at the prime 2, we can show quick collapsing of the AHSS computing the Morava K-theory of some real Grassmanians: this is a non-equivariant result.

An early result in this area was by Neil Strickland. My own contributions have included joint work with Chris Lloyd and also William Balderrama.

In my talk, I'll try to give an overview of some of this.

(joint work with Luca Pol)

Let U be the category of finite groups and conjugacy classes of surjective homomorphisms, or some reasonable subcategory of that. Let A be the category of contravariant functors from U to rational vector spaces (which is equivalent to a certain category of globally equivariant spectra with rational homotopy groups).  The category A has some unusual properties: there is a good theory of duality but finitely generated projective objects are not strongly dualisable, all projective objects are injective but not vice-versa, and so on.  This makes it difficult to analyse the Balmer spectrum of the associated derived category, but we will explain some progress towards that goal.

In recent years work by Calmés-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle have moved the theory of Hermitian K-theory into the framework of stable infinity-categories. I will introduce the basic ideas and notions of this new theory, but as it is often the case when working with K-theory in any form, this can be very hard to describe. I will therefore introduce a tool which might make our life a bit easier: Real Topological Hochschild Homology. I will explain the ingredients that goes into constructing in particular the geometric fixed points of this as a functor, generalising the formula for ring spectra with anti-involution of Dotto-Moi-Patchkoria-Reeh.

I will explain a version of bar-cobar (or "Koszul") duality between covariant and contravariant functors on a category of trees, the proof of which is elementary and explicit. The (known) duality for linear operads is a special case, as is the (new) extension to linear infinity-operads.

I'll introduce some model categories of Cirici, Egas Santander, Livernet and Whitehouse on the categories of filtered chain complexes and bicomplexes (as well as some newer intermediary ones indexed by finite non-empty subsets $S$ of the naturals). Their weak equivalences are determined as isomorphisms on the $(r+1)$-page of the associated spectral sequences where $r = \max S$. I'll show that these are all Quillen equivalent via a zig-zag of totalisation and shift-décalage adjunctions so they all present the same homotopy category. I'll also demonstrate the model structures of filtered chains are in fact monoidal model categories satisfying the monoid axiom. By a result of Shipley and Schwede, we then obtain model structures of filtered differential graded algebras with the same weak equivalences enhancing previous work of Halperin and Tanré.

The complete graph operad is an E_n-operad, completely combinatorial in nature, and apparently occupying a central position in the world of E_n-operads. This in spite of the fact that up to now there seems to be no (correct) proof in the literature that this operad actually is E_n.  I'll discuss some aspects of this operad that I didn't get to in my crash course last spring, but I will try to make the talk independent of what was discussed in that course.

I'll discuss recent joint work with Muriel Livernet. We consider the homotopy theory of the category of spectral sequences with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.

The notion of convexity of sets can be captured in a category theoretic way using a what is known as a monad which associates to a space the finite formal convex combinations of elements. Various authors have looked at such convexity monads on categories of metric spaces. It became clear to me that the work of Fritz-Perrone on this could be naturally expressed if you considered metric spaces as enriched categories, that is categories enriched over a category non-negative real numbers.

In this talk I'll explain this point of view and how notions of concave and convex maps naturally arise when you think higher-categorically. The work is motivated by an attempt to combine two categorical approaches to thermodynamics, one from Lawvere involving enriched categories and one from Baez-Lynch-Moeller involving convexity; I might mention some aspects of that if time permits.

An assembly map is a universal approximation of a homotopy-invariant functor by a generalised homology. In this talk, we introduce the concept and examine examples. When we have an assembly map, we have an associated generalised Novikov conjecture, stating that the map is injective when applied to the classifying space of a group. The plan is to show a general technique coming from coarse geometry to prove injectivity of the assembly map for certain classes of groups.

PROPs are "product and permutation categories". They encode structure borne by objects in a symmetric monoidal category. In this talk I will discuss how the PROP that indexes the structure of a monoid in a symmetric monoidal category is closely related to the theory of crossed simplicial groups. I will then report on recent work (and work in progress) which generalizes this in two ways. I will discuss, firstly, how we can extend known results in the symmetric case to cover monoids with extra structure and, secondly, how we can translate all the results to the setting of braided monoidal categories.

Analogies between low-dimensional topology and number theory have been suggested for over a century. One thing I am interested in at the moment is seeing how we can use the algebra of racks and quandles to classify such objects and understand their symmetries. In this talk, I will briefly introduce this algebra, sketch my work in progress, and indicate some possible future directions if time permits.