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.Comment: 29 pages, 3 figures; some small correction

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Finite rigid sets and homologically non-trivial spheres in the curve complex of a surface

Aramayona and Leininger have provided a "finite rigid subset" $\mathfrak{X}(\Sigma)$ of the curve complex $\mathscr{C}(\Sigma)$ of a surface $\Sigma = \Sigma^n_g$, characterized by the fact that any simplicial injection $\mathfrak{X}(\Sigma) \to \mathscr{C}(\Sigma)$ is induced by a unique element of the mapping class group $\mathrm{Mod}(\Sigma)$. In this paper we prove that, in the case of the sphere with $n\geq 5$ marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a $\mathrm{Mod}(\Sigma)$-module generator for the reduced homology of the curve complex $\mathscr{C}(\Sigma)$, answering in the affirmative a question posed by Aramayona and Leininger. For the surface $\Sigma = \Sigma_g^n$ with $g\geq 3$ and $n\in \{0,1\}$ we find that the finite rigid set $\mathfrak{X}(\Sigma)$ of Aramayona and Leininger contains a proper subcomplex $X(\Sigma)$ whose reduced homology class is a $\mathrm{Mod}(\Sigma)$-module generator for the reduced homology of $\mathscr{C}(\Sigma)$ but which is not itself rigid.Comment: 21 pages, 7 figures; Section 4 revised along with minor corrections throughou

The Casson invariant and the word metric on the Torelli group

We bound the value of the Casson invariant of any integral homology 3-sphere $M$ by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group \T, of the element of \T associated to any Heegaard splitting of $M$. We construct examples which show this bound is asymptotically sharp.Comment: 5 pages, minor corrections; to appear in C. R. Math. Acad. Sci. Pari

The mapping class group of connect sums of S2×S1S^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 $(\mathbb{Z}/2)^n$.Comment: 18 pages, 2 figures; minor revisio

Noncyclic covers of knot complements

Hempel has shown that the fundamental groups of knot complements are residually finite. This implies that every nontrivial knot must have a finite-sheeted, noncyclic cover. We give an explicit bound, $\Phi (c)$, such that if $K$ is a nontrivial knot in the three-sphere with a diagram with $c$ crossings and a particularly simple JSJ decomposition then the complement of $K$ has a finite-sheeted, noncyclic cover with at most $\Phi (c)$ sheets.Comment: 29 pages, 8 figures, from Ph.D. thesis at Columbia University; Acknowledgments added; Content correcte