Thomas Church

FI-modules and stability for representations of symmetric groups

In this paper we introduce and develop the theory of FI-modules. We apply this theory to obtain new theorems about: - the cohomology of the configuration space of n distinct ordered points on an arbitrary (connected, oriented) manifold - the diagonal coinvariant algebra on r sets of n variables - the cohomology and tautological ring of the moduli space of n-pointed curves - the space of polynomials on rank varieties of n x n matrices - the subalgebra of the cohomology of the genus n Torelli group generated by H^1 and more. The symmetric group S_n acts on each of these vector spaces. In most cases almost nothing is known about the characters of these representations, or even their dimensions. We prove that in each fixed degree the character is given, for n large enough, by a polynomial in the cycle-counting functions that is independent of n. In particular, the dimension is eventually a polynomial in n. In this framework, representation stability (in the sense of Church-Farb) for a sequence of S_n-representations is converted to a finite generation property for a single FI-module.

FI-modules over Noetherian rings

FI-modules were introduced by the first three authors to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FI-module implies representation stability for the corresponding sequence of Sn ‐representations. In this paper we prove the Noetherian property for FI-modules over arbitrary Noetherian rings: any sub-FI-module of a finitely generated FI-module is finitely generated. This lets us extend many results to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman’s central stability for homology of congruence subgroups. 20B30; 20C32

Homological stability for configuration spaces of manifolds

Let Cn(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups Hi(Cn(M);ℚ) are representation stable in the sense of Church and Farb (arXiv:1008.1368). Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space Bn(M) satisfies classical homological stability: for each i, Hi(Bn(M);ℚ)≈Hi(Bn+1(M);ℚ) for n>i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural examples where rational homological stability holds even though integral homological stability fails.To prove the main theorem, we introduce the notion of monotonicity for a sequence of Sn-representations, which is of independent interest. Monotonicity provides a new mechanism for proving representation stability using spectral sequences. The key technical point in the main theorem is that certain sequences of induced representations are monotone.

Homology of FI-modules

We prove an explicit and sharp upper bound for the Castelnuovo-Mumford regularity of an FI-module V in terms of the degrees of its generators and relations. We use this to refine a result of Putman on the stability of homology of congruence subgroups, extending his theorem to previously excluded small characteristics and to integral homology while maintaining explicit bounds for the stable range.

On cap sets and the group-theoretic approach to matrix multiplication

In 2003, Cohn and Umans described a framework for proving upper bounds on the exponent ω of matrix multiplication by reducing matrix multiplication to group algebra multiplication, and in 2005 Cohn, Kleinberg, Szegedy, and Umans proposed specific conjectures for how to obtain ω = 2. In this paper we rule out obtaining ω = 2 in this framework from abelian groups of bounded exponent. To do this we bound the size of tricolored sum-free sets in such groups, extending the breakthrough results of Croot, Lev, Pach, Ellenberg, and Gijswijt on cap sets. As a byproduct of our proof, we show that a variant of tensor rank due to Tao gives a quantitative understanding of the notion of unstable tensor from geometric invariant theory.

Representation stability in cohomology and asymptotics for families of varieties over finite fields

We consider two families X_n of varieties on which the symmetric group S_n acts: the configuration space of n points in C and the space of n linearly independent lines in C^n. Given an irreducible S_n-representation V, one can ask how the multiplicity of V in the cohomology groups H*(X_n;Q) varies with n. We explain how the Grothendieck-Lefschetz Fixed Point Theorem converts a formula for this multiplicity to a formula for the number of polynomials over F_q (or maximal tori in GL_n(F_q), respectively) with specified properties related to V. In particular, we explain how representation stability in cohomology, in the sense of [CF, arXiv:1008.1368] and [CEF, arXiv:1204.4533], corresponds to asymptotic stability of various point counts as n goes to infinity.

Generating the Johnson filtration

For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, there exists some G_k >= 0 such that Torelli_g^1(k) is generated by elements which are supported on subsurfaces whose genus is at most G_k. We also prove similar theorems for the Johnson filtration of Aut(F_n) and for certain mod-p analogues of the Johnson filtrations of both the mapping class group and of Aut(F_n). The main tools used in the proofs are the related theories of FI-modules (due to the first author together with Ellenberg and Farb) and central stability (due to the second author), both of which concern the representation theory of the symmetric groups over Z.

Orbits of curves under the Johnson kernel

This paper has two main goals. First, we give a complete, explicit, and computable solution to the problem of when two simple closed curves on a surface are equivalent under the Johnson kernel. Second, we show that the Johnson filtration and the Johnson homomorphism can be defined intrinsically on subsurfaces and prove that both are functorial under inclusions of subsurfaces. The key point is that the latter reduces the former to a finite computation, which can be carried out by hand. In particular this solves the conjugacy problem in the Johnson kernel for separating twists. Using a theorem of Putman, we compute the first Betti number of the Torelli group of a subsurface.

Parameterized Abel–Jacobi maps and abelian cycles in the Torelli group

Let Ig,� denote the Torelli group of the genus g � 2 surface Sg with one marked point. This is the group of homotopy classes (rel basepoint) of homeomorphisms of Sg fixing the basepoint and acting trivially on H := H1(Sg;Q). In 1983 Johnson constructed a beautiful family of invariants �i: Hi(Ig,�;Q) ! V i+2 H for 0 � i � 2g − 2, using a kind of Abel–Jacobi map for families. He used these invariants to detect nontrivial cycles in Ig,�. Johnson proved that �1 is an isomor��

A stability conjecture for the unstable cohomology of SL_n Z, mapping class groups, and Aut(F_n)

In this paper we conjecture the stability and vanishing of a large piece of the unstable rational cohomology of SL_n Z, of mapping class groups, and of Aut(F_n).