The panted cobordism group of cusped hyperbolic 3-manifolds

TL;DR

This paper investigates the panted cobordism group of cusped hyperbolic 3-manifolds, establishing an isomorphism with the first homology of the special orthogonal bundle for certain parameters.

Contribution

It introduces a modified panted cobordism group and proves its isomorphism to a topological invariant of the manifold, connecting geometric and algebraic structures.

Findings

The modified panted cobordism group is isomorphic to H_1(SO(M); Z) under certain conditions.

The study extends understanding of geometric structures in hyperbolic 3-manifolds.

Provides a new algebraic-topological characterization of cusped hyperbolic 3-manifolds.

Abstract

For any oriented cusped hyperbolic $3$-manifold $M$, we study its $(R,\epsilon)$-panted cobordism group, which is the abelian group generated by $(R,\epsilon)$-good curves in $M$ modulo the oriented boundaries of $(R,\epsilon)$-good pants. In particular, we prove that for sufficiently small $\epsilon>0$ and sufficiently large $R>0$, some modified version of the $(R,\epsilon)$-panted cobordism group of $M$ is isomorphic to $H_1(\text{SO}(M);\mathbb{Z})$.