Torsions of 3-dimensional manifolds

Three-dimensional topology encompasses the study of geometric structures and topological invariants of 3-manifolds and knots. This work focuses on the invariant known as maximal abelian torsion, denoted T, applicable to compact smooth or piecewise-linear manifolds and finite CW-complexes X. The torsion T(X) is an element of an extension of the group ring Z[Hl(X)] and can be analyzed within the context of simple homotopy theory. It remains invariant under simple homotopy equivalences, distinguishing homotopy equivalent but non-homeomorphic CW-spaces and manifolds, such as lens spaces. Additionally, T can differentiate orientations and Euler structures. The significance of torsion T lies in its crucial role in three-dimensional topology. It is closely linked to several fundamental topological invariants of 3-manifolds. Specifically, the torsion T(M) of a closed oriented 3-manifold M determines the first elementary ideal of the fundamental group π1(M) and the Alexander polynomial of π1(M). Furthermore, T(M) is associated with the cohomology rings of M with coefficients in Z and Z/rZ (for r ≥ 2), the linking form on Tors Hi(M), Massey products in the cohomology of M, and the Thurston norm on H2(M).