Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1110.0876

Generating the Torelli group

Allen Hatcher·Dan Margalit
View original source ↗Read source document ↗Export metadata ↓

Abstract

We give a new proof of the theorem of Birman-Powell that the Torelli subgroup of the mapping class group of a closed orientable surface of genus at least 3 is generated by simple homeomorphisms known as bounding pair maps. The key ingredient is a proof that the subcomplex of the curve complex of the surface spanned by curves within a fixed homology class is connected.

Explore related subjects

math.GTmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Allen Hatcher, Dan Margalit. 2012-02-27. Generating the Torelli group. https://arxiv.org/abs/1110.0876

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Chern-Simons invariants and volumes of representations in Nil, Sol, and Euclidean geometries

In this paper, we realize volumes of representations as real-valued Chern-Simons invariants in Nil, Sol, and Euclidean geometries. To this end, we formulate a Chern-Simons invariant of a pair of connections on a principal bundle that need not be trivial. For a connected closed oriented 3-manifold $M$ and a representation $ρ\colonπ_1(M)\to G$ into the identity component $G$ of the isometry group of one of these geometries, we construct an auxiliary connection on the associated flat $G$-bundle. We show that, for a suitably normalized invariant polynomial, the Chern-Simons invariant of the auxiliary and flat connections equals the volume of the representation. For the holonomy representation of a geometric structure, this invariant recovers the Riemannian volume. We also compute the Chern-Simons invariant of the Levi-Civita connection for representative closed manifolds in each of these geometries.

math.GT↗
arXiv2026

Khovanov homology and refined bounds for Gordian distances

From Khovanov homology, we extract a new lower bound for the Gordian distance of knots, which combines and strengthens the previously existing bounds coming from Rasmussen invariants and from torsion invariants. We also improve the bounds for the proper rational Gordian distance.

math.GT↗
arXiv2026

From arcs to curves: quadratic growth of 1-systems

We show that a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $χ$ has at most $2016|χ|^2+338|χ|$ curves. Up to multiplicative constants, this resolves a thirty-year old problem (see Problem 2.12(b) from the K3 Problem List). Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of tulips, flowers, and stem systems in order to account for how certain polygons built from pairs of curves in the collection distribute area over the surface.

math.GT↗