Topological Invariants of 3-Manifolds with Boundary

TL;DR

This paper explores the topological invariants of 3-manifolds with boundary, detailing handle decompositions, fundamental groups, and homology groups for various specific examples including knot complements.

Contribution

It provides detailed calculations and descriptions of topological invariants for a range of 3-manifolds, including explicit examples like knot complements and classical spaces.

Findings

Computed handle decompositions for specific 3-manifolds

Determined fundamental groups and homology groups for examples

Extended results to include some knot complements

Abstract

This paper presents, with explanatory details, the handle decompositions, fundamental groups and homology groups of 3-manifolds, including some knot complements. Hence, along this paper, when the word manifold appears it is implicit that its dimension is 3, except when explicitly generalized for n dimensions, n natural. The results were obtained for: 3-torus, projective space P^3, trefoil (3^1), figure-eight (4^1), cinquefoil (5^1) and three-twist (5^2).