IMUNAM Algebraic Topology Seminar

Una co-álgebra es una estructura dual a la de un álgebra: un espacio vectorial equipado con un co-producto que satisface ciertas propiedades.

Una co-álgebra simplicial es una colección de co-álgebras C₀, C₁, C₂,… con los datos adicionales de ciertos mapas de entre estas que satisfacen las ecuaciones simpliciales. Un ejemplo de una co-álgebra simplicial es la linearización de un conjunto simplicial con coproductos inducidos por el mapa diagonal. En esta plática discutiremos diferentes “teorías homotópicas” para co-álgebras desde el punto de vista de categorías de modelos. Discutiremos el significado de cada una de estas teorías como modelos co-algebraicos para espacios topológicos (o conjuntos simpliciales) salvo distintas nociones de equivalencia débil. La plática estará basada en trabajo en progreso con George Raptis.

Una vieja pregunta de H. Rosenberg, tratada por J.-C. Yoccoz en su tesis, trata sobre la conexidad por arcos del espacio de los difeomorfismos que conmutan del círculo. Hasta el día de hoy ésta sigue en abierto, salvo en casos de muy baja regularidad (continua y C¹). En esta charla presentaré un trabajo en colaboración con Hélène Eynard en que probamos un teorema de conexidad por arcos para los difeomorfimos conmutantes del intervalo en una regularidad ligeramente superior.

En esta plática estudiamos la monodromía de cierta familia de superficies K3 complejas realizadas como cubiertas ramificadas sobre curvas planares cuárticas. La monodromía de la familia universal de dichas curvas cuárticas ha sido determinada por Jansen, y tiene una relación estrecha con el grupo Sp(6, 2) de automorfismos de las 28 bitangentes a una curva cuártica. En esta charla hablaré de este y varios ejemplos de otras familias universales de hipersuperficies suaves. Daremos el cálculo explícito para la familia de las superficies en cuestión, usando una superficie de del Pezzo de grado 2 como cubierta intermedia auxiliar, así como resultados de Kondo y Allcock, Carlson, Toledo sobre la descripción del espacio moduli de curvas cuárticas como cociente de una 6-bola compleja.

Recent work of Boyd, Hepworth and Patzt studies stability patterns in the homology of sequences of abstract algebras. Inspired by their ideas, we introduce the “cellular Davis complex” of a Temperley-Lieb algebra. This contractible chain complex admits a simple diagrammatic description and can be seen as an algebraic analogue of the Davis complex of a Coxeter group. Extending previous results of Boyd and Hepworth, we will explain how this chain complex can be used to show that the homology of any Temperley-Lieb algebra on an odd number of strands vanishes in positive degrees.

An isovariant map between G spaces is an equivariant map that preserves isotropy groups. In this talk, I will discuss how to define a homotopy theory for the category of G spaces with isovariant maps. In this model structure, G manifolds behave very well, and we will discuss some applications to the study of manifolds. This is joint work with Inbar Klang.

The parametrized approach to motion planning offers flexibility for variable situations, typically encoded in a fiber bundle with the base space parametrizing external constraints on the system. Here, the input (initial and terminal states) and the output (a path between them) of a motion planning algorithm must all be subject to the same external conditions (that is, they are in the same fiber of the bundle). We consider this parametrized motion planning problem where each of the spaces involved is the complement of a union of hyperplanes in a complex vector space. Under a combinatorial hypothesis of supersolvability, we determine the parametrized topological complexity of a fiber bundle of arrangement complements. This is joint work with Dan Cohen.

Bicategorical shadows, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. I’ll begin the talk by reviewing this framework and introducing the equivariant Hochschild invariants we will discuss, twisted THH and Hochschild homology for Green functors. I will then talk about joint work with Adamyk, Gerhardt, Hess, and Kong, in which we prove that these equivariant Hochschild invariants are bicategorical shadows, from which we deduce that they satisfy Morita invariance. We also show that they receive trace maps from an equivariant version of algebraic K-theory.

Let X be a projective variety, and C be an algebraic curve. The topological problem of computing the homology of the space of algebraic maps from C to X, is analogous to the arithmetic problem of counting rational points on X. I will talk about the history of this problem, its connection to the topology of loop spaces, and joint work with Ronno Das considering the case where X is a del Pezzo surface (or more generally a blowup of projective space at a finite set of points).

Special day, time and length: Tuesday, 4pm–5:30pm

Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary stability for unordered configuration spaces of manifolds. The main difficulty is the compact case (the non-compact case was previously known by some experts). In the compact case, there are no obvious stabilization maps and the homology does not stabilize but is periodic. We resolve this issue by constructing a chain-level stabilization map for configuration spaces of compact manifolds.

The absolute Galois group of the rationals \(\mathsf{Gal}(\mathbb{Q})\) is one of the most important concepts in number theory. Although we cannot explicitly describe more than two elements in this infinite group, we know it acts on well-known algebraic and topological objects in compatible ways. Grothendieck-Teichmüller theory uses these representations to study this Galois group. One of the most important representations comes from the compatible actions of \(\mathsf{Gal}(\mathbb{Q})\) on all the profinite mapping class groups of surfaces. In this talk, we introduce an algebraic tool called an infinity modular operad and use it to construct an infinity modular operad of surfaces capturing the compatibility structure above. We show this admits an action of \(\mathsf{Gal}(\mathbb{Q})\), translating the Grothendieck-Teichmüller program into the theory of infinity modular operads, which provides new ideas and tools to approach this problem. This is joint work with Marcy Robertson.

In this talk we will consider the topological space of \(n\)-tuples of pairwise commuting elements in a Lie group \(G\). This space arises as a moduli space of flat connections on principal \(G\)-bundles over the \(n\)-torus. We will describe homotopy and homology calculations. One of our main results is the description of the second homotopy group of the space of commuting pairs in any compact Lie group. The connection with gauge theory will be discussed. This is joint work with Alejandro Adem and José Manuel Gómez.

I will discuss a remarkable generalization of Mather’s theorem by Thurston that relates the identity component of diffeomorphism groups to the classifying space of Haefliger structures. The homotopy type of this classifying space plays a fundamental role in foliation theory. However, it is notoriously difficult to determine its homotopy groups. Mather-Thurston theory relates the homology of diffeomorphism groups to these homotopy groups. Hence, this h-principle type theorem has been used as the main tool to get at the homotopy groups of Haefliger spaces. We talk about generalizing Thurston’s method to prove analogue of MT for other subgroups of diffeomorphism groups that was conjectured to hold. Most h-principle methods use the local statement about \(M=\mathbb{R}^n\) to prove a statement about compact manifolds. But Thurston’s method is intrinsically compactly supported method and it is suitable when the local statement for \(M=\mathbb{R}^n\) is hard to prove. As we also shall explain Thurston’s point of view on this “local to global” method implies non abelian Poincare duality.

We discuss joint work with Soren Galatius and Oscar Randal-Williams on the application of higher-algebraic techniques to classical questions about the homology of mapping class groups. This uses a new “multiplicative” approach to homological stability — in contrast to the “additive” one due to Quillen — which has the advantage of providing information outside of the stable range.

The Goodwillie functor calculus tower is an approximation of a homotopy functor which resembles the Taylor series approximation of a function in ordinary calculus. In 2017, B., Johnson, Osborne, Tebbe and Riehl (BJORT, collectively) showed that the directional derivative for functors of an abelian category are an example of a categorical derivative in the sense of Blute, Cockett and Seely. The BJORT result relied on the fact that the target and source of the functors in question were both abelian categories. This leads one to the question of whether or not other sorts of homotopy functors have a similar structure. To address this question, Burke and Ching and I instead use the notion tangent categories, due to Rosicky, Cockett-Cruttwell and via an incarnation due to Leung. The structure of a tangent category is highly reminiscent of the structure of a tangent bundle on a manifold. Indeed, the category of smooth manifolds is a primary and motivating example of a tangent category. In recent work with Burke and Ching, we make precise the notion of a tangent infinity category, and show that the directional derivative for homotopy functors appears as the associated categorical derivative of a particular tangent infinity category. This ties together Lurie’s tangent bundle construction to the categorical literature on tangent categories. In this talk, I aim to explain the categorical notions of differentiation and tangent categories, and explain their relationship to Goodwillie’s functor calculus.

Many sequences of groups and spaces satisfy a phenomenon called “homological stability”. I will present joint work with Hepworth, in which we abstract this notion to sequences of algebras, and prove homological stability for the sequence of Temperley-Lieb algebras. The proof uses a new technique of “inductive resolutions”, to overcome the lack of flatness of the Temperley-Lieb algebras. I will also introduce the “complex of planar injective words” which plays a key role in our work. Time permitting, I will explore some connections to representation theory and combinatorics that arose from our work. I will aim this talk at a broad topological audience, and assume no prior knowledge of homological stability or Temperley-Lieb algebras.

In 1960s and ’70s five separate teams of authors showed that the Eilenberg–Moore spectral sequence computing the cohomology of a homogeneous space collapsed, but there was no word on the ring structure until 2019, when Franz showed it was what one would hope. In this talk we generalize this proof to biquotients \(K\,\text{\\} G / H\), an attractive class of smooth manifolds that contains exotic spheres and essentially all known examples of manifolds admitting a Riemannian metric of nonnegative sectional curvature. The key new ingredient is a natural multiplication on the two-sided bar construction \(B(M,A,N)\) of differential graded algebras, subject to a homotopy-commutativity condition. Background will be explained and proofs limply gestured at. The mod 2 reduction of the formula for the product was proposed by Franz.

The classical scissors congruence problem asks whether given two polyhedra with the same volume, one can cut one into a finite number of smaller polyhedra and reassemble these to form the other. There is an analogous definition of an SK (German “schneiden und kleben,” cut and paste) relation for manifolds and classically defined scissors congruence (SK) groups for manifolds. Recent work of Jonathan Campbell and Inna Zakharevich has focused on building machinery for studying scissors congruence problems via algebraic K-theory, and applying these tools to studying the Grothendieck ring of varieties. I will talk about a new application of this framework: we will construct a K-theory spectrum of manifolds, which lifts the classical SK group, and a derived version of the Euler characteristic. This is joint work with Hoekzema, Semikina, Rovi, and Wells.

Fix a class function of the symmetric group \(S_n\), i.e. a function on the set of partitions of n. For any monic polynomial of degree \(n\) over a fixed finite field the degrees of its irreducible factors will form such a partition and we can ask about the average value of the class function evaluated on all such polynomials. Trevor Hyde (’18) proved by direct calculation that the answer can be expressed in terms of the \(S_n\)-action on the cohomology of the configuration space of \(n\) points in \(\mathbb{R}^3\) (!), but his argument gave no geometric reason for such a formula to exist. We give a geometric proof of Hyde’s formula by applying the Grothendieck–Lefschetz trace formula to the cohomology of a certain highly nonseparated algebraic space obtained by gluing together complements of hyperplanes in the braid arrangement. (Joint with P. Tosteson.)

In the 70s, Fred Cohen and Peter May gave a description of the mod p homology of a free \(E_n\)-algebra in terms of certain homology operations, known as Dyer-Lashof operations and the Browder bracket. These operations capture the failure of the \(E_n\) multiplication to be strictly commutative, and they prove useful for computations. After reviewing the main ideas from May and Cohen’s work, I will discuss a framework to generalize these operations to homology with certain twisted coefficient systems and give a complete classification of twisted operations for \(E_\infty\)-algebras. I will also explain computational results that show the existence of new operations for \(E_2\)-algebras.

El espacio móduli \(\mathsf{BDiff}(M)\) de una variedad suave \(M\) es el espacio que clasifica haces fibrados con fibra \(M\). Entender su tipo de homotopía es entonces la tarea que tiene todo topólogo interesado en clasificar haces fibrados. Para bien o para mal, estos espacios tienden a ser bastante complicados, por lo cual nos toca conformarnos con tratar de calcular algunos de sus invariantes topológicos. Esto tampoco es tarea fácil, incluso en los casos más simples (eg. \(M\) una esfera) son pocas las cuentas que hay disponibles. Sin embargo, hay algo un poco más “cualitativo” que sí se puede afirmar en mucha generalidad: si \(M\) es una variedad suave compacta conexa de dimensión par mayor que 5 y con grupo fundamental finito, entonces todos los grupos de homotopía y homología del espacio móduli \(\mathsf{BDiff}(M)\) son finitamente generados. En esta charla voy a discutir algunas de las ideas que M. Krannich, A. Kupers y yo usamos en la demostración de ésta afirmación.

Power’s 2-categorical pasting theorem, asserting that any pasting diagram in a 2-category has a unique composite, is at the basis of the 2-categorical graphical calculus, which is used extensively to develop the theory of 2-categories. In this talk we discuss an (∞, 2)-categorical analog of the pasting theorem, asserting that the space of composites of any pasting diagram in an (∞, 2)-category is contractible. This result, which is joint with Hackney—Ozornova—Riehl, rediscovers independent work by Columbus.

Adem, Lueck, and their collaborators have shown that in various situations, the Lyndon-Hochschild-Serre spectral sequence associate to a crystallographic group collapses at the second page. Crystallographic groups are built from integral representations finite groups. We describe a generalized notion of induction, which we call dynamical induction, originating in the theory of \(C^\ast\)-algebras, and we show that these collapse results are preserved under induction in an appropriate sense. In particular, we show that the spectral sequence associated to a semidirect product of the form \(\mathbb{Z}^n \rtimes \mathbb{Q}\) collapses whenever the action of \(\mathbb{Q}\) on \(\mathbb{Z}^n\) is induced up from an action of a group of square-free order. This is joint work with my recent Ph.D. student, Chris Neuffer.

Classifying all integral representations of finite groups is essentially impossible. In this talk, we will introduce an integral version of the stable module category for a finite group G and then explain how to use it to give a ’generic’ classification of integral G-representations. Our results globalize the modular case established by Benson, Iyengar, and Krause and relies on the notion of stratification in tensor triangular geometry developed in joint work with Heard and Sanders. Time permitting, I will discuss some further applications.

While collapsibility of CW complexes dates back to the 1930s, collapsibility of directed complexes has not been well studied to date. We define a notion of directed collapsibility in the setting of a directed Euclidean cubical complex that builds on the classical definition of collapsibility. We call this type of collapse a link-preserving directed collapse. In the undirected setting, pairs of cells are removed from a space if a deformation retract remains. In the directed setting, topological properties — in particular, properties of spaces of directed paths — would not always be preserved. The direction of the space can be taken into account by requiring that the past links of vertices remain homotopy equivalent after collapsing. We show that there are computationally simple conditions which preserve the topology of past links. Furthermore, we give conditions for when link-preserving directed collapses preserve the contractability and connectedness of spaces of directed paths. Our results have applications to speeding up the verification process of concurrent programming and to understanding partial executions in concurrent programs. This is joint work with Robin Belton, Robyn Brooks, Stefania Ebli, Lisbeth Fajstrup, Brittany Terese Fasy, and Nicole Sanderson.

This is joint work with Jeremy Miller and Jennifer Wilson. Let \(M\) be an orientable surface of genus \(g\) with one boundary curve, and let \(F_n(M)\) denote the configuration space of \(n\) ordered points in \(M\). The action of \(\mathsf{Homeo}(M,\partial M)\) on \(F_n(M)\) descends to an action of the mapping class group \(\Gamma(M,\partial M)\) on the homology \(H_\ast(F_n(M))\). Our main result is that, for all \(n,i \ge 0\), the i-th stage \(J(i)\) of the Johnson’s filtration of \(\Gamma(M,\partial M)\) acts trivially on \(H_i(F_n(M))\). This extends previous work of Moriyama on certain relative configuration spaces. I will recall the necessary definitions and give a sketch of the proof of the main theorem: the main inputs are Moriyama’s work and a cell stratification of \(F_n(M)\) à la Fox-Neuwirth-Fuchs. I will also present some examples of non-trivial actions of mapping classes in \(J(i-1)\) on elements of \(H_i(F_n(M))\), for small values of \(i\).

Computing the cohomology of arithmetic groups is a fundamental and often difficult problem at the intersection of topology, group theory and number theory. In this talk, I will explain how one can use duality phenoma to compute the rational cohomology of the arithmetic groups \(SL_n(\mathbb{Z})\) and \(Sp_{2n}(\mathbb{Z})\) in “high” dimensions, i.e. close to their virtual cohomological dimension. Specifically, I will talk about joint work with Miller-Patzt-Sroka-Wilson in which we show that \(H^{n(n-1)/2 - 2}(SL_n(\mathbb{Z}); \mathbb{Q}) = 0\) for \(n>3\). This was previously unknown, but confirms a conjecture of Church-Farb-Putman. In ongoing work with Patzt-Sroka, we are also trying to adapt these techniques to the group \(Sp_{2n}(Z)\).

Given a fibre bundle \(p\colon E \to B\) of topological manifolds with \(d\)-dimensional fibres, one might ask whether \(p\) is fibrewise homeomorphic to a smooth bundle. If \(d \ne 4\), then there is a strategy to answer this question based on smoothing and obstruction theory. The main limiting factor in this approach is that it requires knowledge of the homotopy groups of the topological group of diffeomorphisms of a closed d-disc. The study of these homotopy groups has quite a history in geometric topology and turns out to be related to algebraic K- and L-theory, stable homotopy theory, and the combinatorics of finite graphs. In this talk I will explain the relation between smoothing fibre bundles and the homotopy groups of diffeomorphisms of discs, survey the state of the art in the computation of the latter, and explain aspects of recent joint work with Oscar Randal-Williams.