Purdue Topology Seminar, Spring 2020

Introductory Talk: Coalgebraic tools in algebraic topology

Abstract: Coalgebras and comodules are dual concepts that of algebras, rings and modules. In this talk, we will define the notions and present well-known results and examples. For instance, every Thom space (or spectrum) is a particular kind of comodules over the base space. Spectra with a coalgebra structure also appear in chromatic homotopy theory.

Main Talk: Rigidification of higher coalgebras and comodules

Abstract: A classical result of higher algebra relates homotopy coherent rings in spectra with strictly associative (or commutative) ring spectrum in a particular choice of model of spectra (such as symmetric spectra, orthogonal spectra, etc). This process is called rigidification (or rectification). In this talk, we are interested in the dual questions of rigidification for higher coalgebras and comodules. We will show that in full generality, there is no rigidification of coalgebras in spectra. We will instead focus on the discrete case (i.e. differential graded context) and show new results for comodules and coalgebras.

Topics Exam

Title: The Nakayama functor in representation stability theory
Abstract:
I will explain the construction of the Nakayama functor for FI and VI and why it induces an equivalence from a quotient of a category of finitely generated modules to a category of finite dimensional modules. The proof of the equivalence uses some of the key results for FI-modules and VI-modules. This is a joint work with Liping Li and Changchang Xi.

Title:The Galois action on symplectic K-theory
Abstract:
The classifying spaces of arithmetic groups are often modeled by moduli spaces in algebraic geometry. A notable example of this phenomenon is the moduli space of complex abelian varieties, which models the classifying space of integral symplectic groups. As a consequence, the group homology and algebraic K-theory of symplectic groups carry a natural Galois action. In joint work with Soren Galatius and Akshay Venkatesh we compute this Galois action, and find it to have an interesting universal characterization.

Title: String topology and the configuration space of two points
Abstract:
Given a manifold M, Chas and Sullivan construct a Lie bialgebra structure on the homology of the space of (unparametrized) loops using intresections and self-intersections of loops. We give an algebraic description of this structure under Chen’s isomorphism identifying loop space homology with cyclic homology. More precisely, we construct a homotopy involutive Lie bialgebra structure on cyclic cochains that depends on the partition function of a Chern-Simons type field theory. Moreover, we discuss the (non-)homotopy invariance of that structure and its relation to the configuration space of two points.

There is an introductory lecture at 10:30.

Title: Secondary stability and periodicity for unordered configuration spaces
Abstract: Secondary stability is a stability pattern in a range that homological stability does not hold. The first example of secondary stability is Galatius–Kupers–Randal-Williams results on mapping class groups. We prove secondary stability for unordered configuration spaces of manifolds. The main difficulty is the closed case (the open case was previously known by some experts). In the closed 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 configurations of closed manifolds.

Title: Differential graded Koszul duality: a global approach
Abstract: Koszul duality is a phenomenon that shows up in rational homotopy theory, deformation theory and other subfields of algebra and topology. It exists on various levels: as a correspondence between operads and cooperads, algebras over operads and coalgebras over cooperads and modules and comodules over associative algebras (the latter, simplest, version will be relevant to this talk). Usually some kind of conilpotence is assumed on the `co’ side. In this talk I explain what happens if this condition is dropped; the consequences turn out to be quite dramatic. I will show how this non-conilpotent (or global) version of Koszul duality comes up naturally in the study of derived categories of complex algebraic manifolds and infinity local systems on topological spaces. Time permitting, I will also explain how one can construct a global version of deformation theory for certain deformation problems based on this approach. This is joint work with Ai Guan.

Title: Twisted Dyer-Lashof Operations
Abstract: 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 explain computational results that show the existence of additional operations in the twisted case.

Title: Hodge decomposition of string topology
Abstract: Given a closed oriented manifold X, Chas and Sullivan constructed a Lie bracket on the (reduced) S^1-equivariant homology of the free loop space of X, called the string bracket. Furthermore, if the manifold is simply connected, there is a Hodge type decomposition on the S^1-equivariant homology induced by the n-fold coverings of the circle. It is natural to ask whether the string bracket preserve this decomposition. We provide a positive answer under the additional assumption that X is rationally elliptic. Our argument is based on algebraic models of string topology and analogous operations on Hochschild and cyclic homologies. This is a joint work with Yuri Berest and Ajay Ramadoss.

Title:
The “generating function” of configuration spaces, as a source for explicit formulas and representation stability

Abstract:
As countless examples show, sequences of complicated objects
should be studied all at once via the formalism of generating
functions. In this talk I will apply this point of view to the
homology and combinatorics of (orbit-)configuration spaces: using the
notion of twisted commutative algebras, which categorify exponential
generating functions. With this idea the configuration space
“generating function” factors into an infinite product, whose terms
are surprisingly easy to understand. Beyond the intrinsic aesthetic of
this decomposition and its quantitative consequences, it also gives
rise to representation stability – a notion of homological stability
for sequences of representations of differing groups.

Title: Diffeomorphisms of reducible three manifolds and bordisms of group actions on torus

Abstract: I first talk about the joint work with K. Mann on certain rigidity results on group actions on torus. In particular, we show that if the torus action on itself extends to a C^0 action on a three manifold M that bounds the torus, then M is homeomorphic to the solid torus.

Motivated by this result, I will talk about certain finiteness results about classifying space of reducible three manifolds which is related to an unfinished project by Allen Hatcher.

Title: E_\infty algebras and general linear groups of infinite fields

Abstract: Homology of general linear groups of a field F is calculated by the chain complex R(n) = C_*(BGL_n(F)). There are chain maps from the tensor product of R(n) and R(m) to R(n+m), induced by direct sum of F-vector spaces. These products make R into a bigraded differential graded algebra, which is homotopy commutative, and can moreover be given the structure of an E_\infty algebra. In joint work with Alexander Kupers and Oscar Randal-Williams, we study the structure of this E_\infty algebra when F is infinite. We are led to new results about homology of general linear groups, for instance an extension of a relationship between the relative homology of BGL_n relative to BGL_{n-1} and Milnor K-theory, due to Nesterenko and Suslin, as well as new questions.

[Video Recording]

Title: Hermitian K-theory

Abstract: I will start by introducing the Grothendieck-Witt group associated to a ring and discuss some examples. I will show how it relates to the algebraic K-group and the Witt group. I will then explain that this relation can be refined to fibre sequence of spaces relating the Grothendieck-Witt space to algebraic K-theory and Ranicki’s algebraic L-theory. Finally, I will indicate that the Grothendieck-Witt space is closely related to an algebraic version of the geometric cobordism category, and highlight some similarities between the geometric and algebraic world. All of this is joint work with Calmès, Dotto, Harpaz, Hebstreit, Moi, Nardin, Nikolaus and Steimle.

Title: Singular chains on Lie groups, the Cartan relations and Chern-Weil theory

Abstract: Let G be a simply connected Lie group. The space C(G) of smooth singular chains on G has de structure of a differential graded algebra, with product induced by the Eilenberg-Zilber map. We will describe how the category of (sufficiently smooth) modules over this algebra can be described infinitesimally. More precisely, we will show that the category of modules over C(G) is equivalent to the category of represenations of the differential graded Lie algebra Tg, which is universal for the Cartan relations. This extends the usual correspondence between representations of G and representations of the Lie algebra g. We also prove that in case G is compact, this equivalence of categories can be extended to an A_∞-equivalence of dg-Categories. The main ingredients in the proof are Gugenheim’s A_∞-version of de Rham’s theorem, the non- commutative Weil algebra of Alekseev-Meinrenken, and the Van-Est map. If time permits, I will describe how this construction relates to a version of Chern-Weil theory for ∞-local systems on classifying spaces.
This talk is based on joint work with Alexander Quintero.

Title: André–Quillen homology and homological stability for general linear groups of Euclidean domains

Abstract: I will discuss joint work in progress with Kupers and Patzt on improved stable ranges for homological stability for general linear groups of Euclidean domains. The main ingredient is a vanishing result for E_infinity André–Quillen homology. By work of Galatius–Kupers–Randal-Williams, these André–Quillen homology groups are isomorphic to the equivariant homology of certain posets.

Title: Operation-indexing categories and Grothendieck fibrations

Abstract: It has been known since Segal that various small categories can
be used as blueprints for algebraic structures in homotopy theory,
providing alternatives to operads in such questions as for example delooping.
The examples of those categories include finite sets, ordered sets, n-
ordinals of Batanin and various exit path categories of configuration
spaces, as well as categories of operators of general topological operads.

We would like to offer a definition for such
operation-indexing categories, called weak operads or algebraic patterns,
and how to describe homotopy-algebraic structures over such things via
fibrations of model and higher categories. Depending on time and the
interest of the audience, we may attempt to introduce the notion of a
weak approximation as a means of establishing certain “Morita”-type
equivalences in the world of weak operads.

Title: Pseudoisotopies of discs and algebraic K-theory of the integers

Abstract: 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 in a range up to roughly the dimension. This approach is independent of the usual route, does not involve stabilising the dimension, and is homological in nature.

Title: Computing Hecke Operators for Cohomology of Arithmetic Groups

Abstract: I will describe three projects. The first, which is joint with Avner
Ash and Paul Gunnells, concerns arithmetic subgroups Gamma of G =
SL(4,R). We compute the cohomology of the locally symmetric spaces
Gamma\G/K, focusing on the cuspidal degree H^5. We compute a range of
Hecke operators on this cohomology. We find Galois representations
that appear to be attached to the Hecke eigenclasses, based on the
operators we have computed. We have done this for both non-torsion
and torsion classes. The method is to use a cell complex, the
well-rounded retract, due to Ash. The second project, joint with Bob
MacPherson, is an algorithm for computing the Hecke operators on the
cohomology H^d of Gamma in SL(n,R) for all n and all d. For this we
introduce a new retract, the well-tempered complex. In the third
project, joint with Dylan Galt, we give an algorithm for Hecke
operators on H^d of Gamma in Sp(4,R) for all d. The method is to
define an appropriate subcomplex of the well-tempered complex for
SL(4,R).

Title: The fundamental group and homotopy theory over a field

Abstract: I will explain how the fundamental group of a space, as well as its homology groups with coefficients in all possible local systems of vector spaces, are completely determined by chain level algebraic data. This algebraic data may be packaged as the weak equivalence class of the simplicial cocommutative coalgebra of chains under a notion of weak equivalence which involves adjusting the coalgebra structure through the cobar construction and is stronger than quasi-isomorphism. I will outline a theory, of independent interest, of twisting cochains and twisted tensor products at the level of simplicial coalgebras and algebras, which we used to construct the fundamental bialgebra and the universal cover of an abstract connected simplicial cocommutative coalgebra. This opens up the possibility of using the power of abstract algebraic homotopy theory to study geometric spaces with arbitrary fundamental group.