Algebraic Topology of Certain Sasaki Joins

TL;DR

This paper explores the algebraic topology of certain Sasaki joins, revealing torsion in homology and identifying linking forms, thus advancing understanding of their topological invariants.

Contribution

It provides explicit calculations of fundamental group, homology, and cohomology for joins of circle bundles over surfaces with weighted spheres, highlighting torsion phenomena.

Findings

Presence of torsion in integral homology

Identification of linking forms associated with torsion

Topological structure as lens space bundle over a surface

Abstract

The join construction produces a third Sasaki manifold from two others, and we investigate the algebraic topology of the joins of circle bundles over surfaces of positive genus with weighted three-spheres. Topologically, such a join has the structure of a lens space bundle over a surface. We calculate invariants determined by the fundamental group, the homology, and the cohomology. We find that, in general, there is torsion in the integral homology of the join. The torsion gives rise to two linking forms, and we identify these linking forms.