Volume and topology of bounded and closed hyperbolic 3-manifolds

Jason DeBlois; Peter B. Shalen

Home/Archive/arXiv

arXiv · 0902.0576

Volume and topology of bounded and closed hyperbolic 3-manifolds

Jason DeBlois·Peter B. Shalen

View original source ↗Read source document ↗Export metadata ↓

Abstract

Let N be a compact, orientable hyperbolic 3-manifold with connected, totally geodesic boundary of genus 2. If N has Heegaard genus at least 5, then its volume is greater than 6.89. The proof of this result uses the following dichotomy: either N has a long return path (defined by Kojima-Miyamoto), or N has an embedded, codimension-0 submanifold X with incompressible boundary $T \sqcup \partial N$, where T is the frontier of X in N, which is not a book of I-bundles. As an application of this result, we show that if M is a closed, orientable hyperbolic 3-manifold such that H_1(M;Z_2) has dimension at least 5, and if the image in H^2(M;Z_2) of the cup product map has image of dimension at most 1, then M has volume greater than 3.44.

Explore related subjects

math.GT

Keep this discovery

BibTeX RIS

Jason DeBlois, Peter B. Shalen. 2009-02-03. Volume and topology of bounded and closed hyperbolic 3-manifolds. https://arxiv.org/abs/0902.0576

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

KEEP EXPLORING

Related papers

arXiv2026

Bar cohomology of links: beyond Milnor invariants

We develop bar cohomology of link complements as an invariant of links in homology spheres. In this setting, bar cohomology is a Hopf algebra which is calculable using surfaces and their intersection curves in a link complement. In this first in a sequence of works, we introduce the invariant and show that it defines a canonical subspace of the tensor Hopf algebra, which already encodes information about Milnor's link invariants and provides geometrically significant information beyond them.

math.GT↗

arXiv2026

A counterexample to the wrapping number conjecture

We exhibit an annular knot with wrapping number four whose Kauffman bracket has annular degree at most two. This disproves the wrapping number conjecture.

math.GT↗

arXiv2026

Homological lifts of Arnold invariants $J^-$ and $J^+$

Viro's Euler-integral polynomial $P_C(q)$ and the Lanzat--Polyak quantized-curvature polynomial $I_q(C)$ refine Arnold's invariants $J^-$ and $J^+$ for generic immersed one-component plane curves. We construct homological lifts of both. The bigraded region homology retains the singular homology of every connected Alexander-index region; its graded Euler characteristic is $P_C(q)$. The triply graded smoothing-circle homology is generated by the oriented circles of the orientation-preserving smoothing and decategorifies to the smoothing term in $I_q(C)$. Keeping the actual region summands and the boundary regions of every smoothing circle gives a homological refinement of the oriented smoothing configuration, or Seifert state. An infinite family proves strictness: both polynomial data and the ordinary homological lifts agree, while the component-graded region homology and the branch-decomposed circle homology distinguish every pair. Further constructions recover the full $I_q(C)$ by a vertex complex, realize the local change of its curvature integral by edge homology, and give a canonical two-state homology for unoriented curves. Viro described his Euler-integral formula as an analogue of face state-sum formulas for quantum knot polynomials. Through the categorifications developed here, we obtain one concrete homological face-state-sum model realizing that analogy.

math.GT↗