arXiv subjects

Tara Brendle

Publications and source records attributed to Tara Brendle.

6 recordsLinked to original sources

arXiv2022

The mapping class group of connect sums of $S^2 \times S^1$

Let $M_n$ be the connect sum of $n$ copies of $S^2 \times S^1$. A classical theorem of Laudenbach says that the mapping class group $\text{Mod}(M_n)$ is an extension of $\text{Out}(F_n)$ by a group $(\mathbb{Z}/2)^n$ generated by sphere twists. We prove that this extension splits, so $\text{Mod}(M_n)$ is the semidirect product of $\text{Out}(F_n)$ by $(\mathbb{Z}/2)^n$, which $\text{Out}(F_n)$ acts on via the dual of the natural surjection $\text{Out}(F_n) \rightarrow \text{GL}_n(\mathbb{Z}/2)$. Our splitting takes $\text{Out}(F_n)$ to the subgroup of $\text{Mod}(M_n)$ consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of $M_n$. Our techniques also simplify various aspects of Laudenbach's original proof, including the identification of the twist subgroup with $(\mathbb{Z}/2)^n$.

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arXiv2022

The high-dimensional cohomology of the moduli space of curves with level structures II: punctures and boundary

We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds that are super-exponential in each of three variables: number of punctures, number of boundary components, and genus, generalizing work of Fullarton-Putman. Along the way, we give a simplified account of a theorem of Harer explaining how to relate the homotopy type of the curve complex of a multiply-punctured surface to the curve complex of a once-punctured surface through a process that can be viewed as an analogue of a Birman exact sequence for curve complexes. As an application, we prove upper and lower bounds on the coherent cohomological dimension of the moduli space of curves with marked points. For $g \leq 5$, we compute this coherent cohomological dimension for any number of marked points. In contrast to our bounds on cohomology, when the surface has $n \geq1$ marked points, these bounds turn out to be independent of $n$, and depend only on the genus.

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arXiv2018

Normal subgroups of mapping class groups and the metaconjecture of Ivanov

We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N.V. Ivanov, which asserts that any "sufficiently rich" object associated to a surface has automorphism group isomorphic to the extended mapping class group, for a broad class of such objects. As applications, we show: (1) right-angled Artin groups and surface groups cannot be isomorphic to normal subgroups of mapping class groups containing elements of small support, (2) normal subgroups of distinct mapping class groups cannot be isomorphic if they both have elements of small support, and (3) distinct normal subgroups of the mapping class group with elements of small support are not isomorphic. Our results also suggest a new framework for the classification of normal subgroups of the mapping class group.

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arXiv2014

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t=-1

We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping, and so we obtain analogous generating sets for those. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply connected when curves of compact type are added.

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arXiv2011

Cohomology of the hyperelliptic Torelli group

Let SI(S_g) denote the hyperelliptic Torelli group of a closed surface S_g of genus g. This is the subgroup of the mapping class group of S_g consisting of elements that act trivially on H_1(S_g;Z) and that commute with some fixed hyperelliptic involution of S_g. We prove that the cohomological dimension of SI(S_g) is g-1 when g > 0. We also show that H_g-1(SI(S_g);Z) is infinitely generated when g > 1. In particular, SI(S_3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B_n at t = -1 has cohomological dimension equal to the integer part of n/2, and it has infinitely generated homology in this top dimension.

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arXiv2010

Configuration spaces of rings and wickets

The main result in this paper is that the space of all smooth links in Euclidean 3-space isotopic to the trivial link of n components has the same homotopy type as its finite-dimensional subspace consisting of configurations of n unlinked Euclidean circles (the "rings" in the title). There is also an analogous result for spaces of arcs in upper half-space, with circles replaced by semicircles (the "wickets" in the title). A key part of the proofs is a procedure for greatly reducing the complexity of tangled configurations of rings and wickets. This leads to simple methods for computing presentations for the fundamental groups of these spaces of rings and wickets as well as various interesting subspaces. The wicket spaces are also shown to be K(G,1)'s.

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