Betti numbers in algebra and homotopy theory – Abstracts

Alejandro Adem (Vancouver)
Minimal Euler Characteristics for Even-Dimensional Manifolds with A Given Fundamental Group

If $M$ denotes a closed, orientable even-dimensional manifold with a given fundamental group $G$, then what restriction does this impose on the Euler characteristic of $M$? In the particular case when $\chi(M) = 2$ we have the related problem of determining which finite groups can be the fundamental group of a closed topological $2n$-manifold $M$ with the rational homology of the $2n$-sphere. This is an interesting open question in the case of $4$-manifolds.

In this talk we will discuss estimates for the minimal Euler characteristic of even dimensional manifolds with a given fundamental group and a highly connected universal cover. As an application we obtain new restrictions for non-abelian finite groups to arise as fundamental groups of rational homology $4$–spheres. This is joint work with Ian Hambleton.

Luchezar Avramov (Sofia)
Unpacking Betti tables

The talk concerns (finitely generated graded) modules over a (standard graded commutative) algebra over some field. A few naive, interrelated questions provide a framework for the discussion:

What information on a module can be read off its Betti table?
How to recognize if a matrix of integers is the Betti table of a module?
How to construct a module with a given Betti table?

I will describe environments where precise questions can be posed, answer some of these, and illustrate a few relevant techniques on specific examples.

William Balderrama (Charlottesville)
Degrees of stable maps between compactifications of fixed point free representations of cyclic p-groups

I will describe work that determines the possible degrees on fixed points of equivariant stable maps $S^V \to S^W$ whenever $V$ and $W$ are fixed point free representations of a cyclic $p$-group. For the cyclic group of order $2$, this problem was introduced by Bredon and resolved by Landweber in the 1960s, but little progress has been made since then for larger groups. The Balmer spectrum of finite $C_{p^n}$-spectra suggests that these maps should be related to phenomena surrounding topological K-theory, and I will describe a lift of equivariant Bott periodicity to the equivariant stable stems that realizes this. This talk is based on joint work with Shangjie Zhang and Yueshi Hou.

Petter Andreas Bergh (Trondheim)
Complexes with small homology over finite tensor categories

Our starting point is G. Carlsson's conjecture on group actions on CW-complexes: if an elementary abelian p-group acts freely on such a complex, then the total homology of the complex cannot be too small. The algebraic version of this conjecture says that the same holds for every nontrivial perfect complex over the group algebra of such a group. However, the algebraic version turns out not to hold: in 2018, S. Iyengar and M. Walker showed that when p is odd, then counterexamples always exist for elementary abelian p-groups of rank at least 8. Recently, J. Carlson extended this to arbitrary finite groups having p-rank at least 8, and the topic of this talk is a generalization to finite tensor categories. Namely, we show that if such a category has finitely generated cohomology, and the Krull dimension of its cohomology ring is at least 8, then there exist infinitely many non-isomorphic and nontrivial perfect complexes having small homology.

Ben Briggs (London)
Free loop spaces and the growth of Hochschild homology, or better, cotangent homology

An old problem of Gromov and Vigué-Poirrier predicts that the betti numbers of a free loop space (of a nice enough space) should almost always grow exponentially. In commutative algebra this translates to the following question: if the Hochschild homology groups of a $k$-algebra $A$ grow less than exponentially, must $A$ be locally complete intersection? It can be easier to handle the cotangent homology of $A$, which sits inside Hochschild homology if $k$ has characteristic zero. I'll talk about work in progress with Srikanth Iyengar and Greg Stevenson where we sort out some cases of this.

Natàlia Castellana (Barcelona)
Stratification in equivariant homotopy theory via geometric fixed points

In this project with T. Barthel, D. Heard, N. Naumann and L. Pol, we categorify the classical strong Quillen's stratification theorem for the mod p cohomology of finite groups in the context of equivariant tensor-triangular geometry. The strategy shows how it is controlled by the non-equivariant tensor-triangular geometry of the geometric fixed points.

John Greenlees (Warwick)
Chromatic Smith theory and Balmer spectra of equivariant cohomology theories (after Smith, Floyd, Smith, Kuhn-Lloyd)

One occurence of Betti numbers is associated to Morava K-theories. For a fixed prime $p$, there are homology theories $K(n)$ for $0\leq n \leq \infty$. For $n=0$ this is rational homology. For $n=\infty$ this is mod $p$ ordinary homology, and for all other values of $n$ we have $K(n)_*=\mathbb{F}_p[v_n, v_n^{-1}]$ where $v_n$ is of degree $2(p^n-1)$. In any case, they are all graded fields and we may consider the Chromatic Betti numbers $k_n(X)=\dim (K(n)_*(X))$ for finite complexes $X$.

The talk will describe some fundamental applications of Morava K-theories to equivariant homotopy theory.

For finite groups $G$, we may consider the tensor triangulated category of $G$-equivariant cohomology theories. Its broad structure is controlled by the Balmer spectrum, which is known for all groups $G$ as a set. However the Zariski topology is known for very few groups (including all abelian groups).
The situation is quickly reduced to a single prime $p$, so we assume that $G$ is a $p$-group and all complexes are localized at $p$. The topology on the Balmer spectrum is controlled by the vanishing of Morava K-theories of fixed points. In other words, for subgroups $H$ of $G$, one asks when does $K(n)_*(X^H)$ imply $K(m)_*(X^G)=0$? (If so we say the Chromatic Smith theorem $(G,H,m,n)$ holds. (This is because of the theorem for mod $p$ cohomology proved by PA Smith in 1938 (which states it is a theorem when $m = n = \infty$))

As for Betti numbers, one asks when $k(n)_*(X^H) \geq k(m)_*(X^G)$ for all finite $G$-complexes $X$? (if so, we say the Chromatic Floyd theorem $(G,H,m,n)$ holds (This is because of the theorem for mod $p$ cohomology proved by EE Floyd in 1952 (which states it is a theorem when $m = n = \infty$))). It is obvious that if the Floyd Theorem holds for $(G,H,m,n)$ then so does the Smith Theorem. Kuhn and Lloyd show that Jeff Smith's ideas prove that the reverse implication also holds, which gives new methods to estimate the least $n$ for which $(G,H,m,n)$ Smith Theorem holds.

The talk will describe what is known, largely following the paper Chromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups by N.Kuhn and J.Lloyd.

Eloísa Grifo (Lincoln)
Bounding betti numbers in commutative algebra

This talk is intended as an introduction to the theory of betti numbers in commutative algebra, and to some of the open conjectures on the subject, such as the Buchsbaum--Eisenbud--Horrocks Conjecture. We will talk about what betti numbers are and how to compute them, and we will survey recent results on bounds for Betti numbers of modules, with an emphasis on lower bounds.

Jesper Grodal (Copenhagen)
Homotopical degree of symmetry

Conjectures in algebra and homotopy theory circle around the idea that "small" objects cannot have "large" degree of symmetry. From a homotopical point of view, this can be formulated as bounds on the possible maps from BG to BAut(X), where BG is the classifying space of a finite group G, and BAut(X) is the classifying space of the space of self homotopy equivalences of X. In my talk I'll describe some of what is known and unknown about this question.

Jeremiah Heller (Urbana-Champaign)
Equivariant motivic cohomology over C

For a finite group G, Bredon motivic cohomology is an invariant for smooth G-schemes, over a base field, which mixes motivic cohomology and topological Bredon cohomology. I'll talk about computations of the coefficient rings and Steenrod algebras and operations, and some applications of these computations, over the complex numbers and for G = C_2.
This is joint work with M. Voineagu and P. A. Ostvaer.

Srikanth Iyengar (Salt Lake City)
Modules of finite length and finite projective dimension

Over a noetherian local ring R, how small can a module M of finite projective dimension be? My talk addresses this question, and the measure of size I plan to focus on is the length of M and also its Loewy length. I will present an overview of results, recent and not so recent, due to various authors, including Avramov, Buchweitz, Miller, Ma, Nawaj KC, Pollitz, Soto Levins, Walker, and myself.

Janina Letz (Bielefeld)
Koszul homomorphisms and universal resolutions in local algebra

I will define a Koszul property for a homomorphism of local rings $\varphi \colon Q \to R$. Koszul homomorphisms have good homological properties. Using $\mathrm{A}_\infty$-structures one can construct universal free resolutions of $R$-modules from free resolutions over $Q$, generalizing the classical construction by Priddy. This recovers the resolutions of Shamash and Eisenbud for complete intersection homomorphisms and the resolutions of Iyengar and Burke for Golod homomorphisms. This is based on work with Ben Briggs, James Cameron and Josh Pollitz.

Clover May (Trondheim)
Representations of Mackey functors and Eilenberg–MacLane spectra

An algebra's representation type, tame or wild, determines whether it is possible to describe all its indecomposable representations in a meaningful way. Mackey functors were introduced by Dress and Green to encode operations that behave like restriction and induction in representation theory. They play a central role in equivariant homotopy theory, where homotopy groups are replaced by homotopy Mackey functors. In this talk I will discuss the representation type and derived representation type of cohomological Mackey functors, as well as some consequences for equivariant Eilenberg--MacLane spectra. This is joint work in progress with Jacob Grevstad.

Berrin Şentürk (Ankara)
An algebraic approach to rank conjecture and examples of small rank

A well-known Rank Conjecture asserts that if an elementary abelian $p$-group acts freely and cellularly on a product of spheres, then the rank of the group cannot exceed the number of spheres in the product. When $p = 2$, we have an algebraic version of the conjecture for a differential graded module $M$ over a polynomial ring in variables of degree $−1$. In this talk, we discuss the properties of the differentials of such modules. We state a stronger conjecture concerning the varieties of square-zero strictly upper triangular matrices corresponding to these differentials. By stratifying these varieties via Borel orbits and using the corresponding free flag construction, we show that $(\mathbb{Z}/2\mathbb{Z})^4$ cannot act freely on a product of $3$ spheres of any dimensions.

Keller VandeBogert (Notre Dame)
The Total Rank Conjecture in Characteristic Two

In this talk, I'll go over the steps required to prove the total rank conjecture in characteristic two (based on joint work with Mark Walker). I'll also mention some more recent results that point toward the "correct" formulation of the generalized total rank conjecture, including an unexpected new source of counterexamples arising from classical representation theory. Some of this is also based on joint work with Steven Sam.

Leopold Zoller (Cologne)
On equivariant notions of formality

Formal spaces play an important role in rational homotopy theory, where they are in some sense the simplest spaces with a given cohomology ring. We investigate possible translations of this concept to the realm of torus actions on spaces and discuss connections with the Toral Rank Conjecture.