Topological invariants of 3-dimensional manifold with boundary by using crossed module

TL;DR

This paper extends the use of topological invariants derived from crossed modules, previously applied to closed manifolds, to compact 3-dimensional manifolds with boundaries, including knot complements, enhancing their applicability.

Contribution

It demonstrates how to adapt existing crossed module invariants to 3-manifolds with boundary, broadening their scope beyond closed manifolds.

Findings

Invariants can be applied to manifolds with boundary.

Method for counting correct colors over triangulations.

Extension to knot complements.

Abstract

J.H.C. Whitehead introduced the concept of crossed modules in the early 20th century. These crossed modules are crucial for algebraic models of 2-type homotopy, which involve connected spaces with no higher than second-degree homotopy groups. They consist of two groups and certain relations between them, with known connections to 2-groups. By employing crossed modules, we can develop invariants for closed 3-dimensional and 4-dimensional manifolds. The validity of these invariants was established in a paper authored by F.Girelli, H.Pfeiffer, and E.M.Popescu([4]). Essentially, these invariants involve counting correct colors over the triangulation of a closed manifold. Interestingly, I've discovered that these invariants can also be applied to compact 3-dimensional manifolds with boundaries. Therefore, in this paper, I intend to demonstrate how these invariants can be utilized for compact 3-dimensional manifolds with boundaries, including the complements of knots.