TL;DR
This paper reviews the relationship between manifold triangulations and the three-dimensional homology cobordism group, focusing on local equivalence methods from advanced Floer homology theories.
Contribution
It synthesizes current knowledge on the homology cobordism group using modern Floer homology techniques, highlighting recent methodological advances.
Findings
Overview of the homology cobordism group
Application of Pin(2)-equivariant Seiberg-Witten Floer spectra
Use of involutive Heegaard Floer homology
Abstract
The study of triangulations on manifolds is closely related to understanding the three-dimensional homology cobordism group. We review here what is known about this group, with an emphasis on the local equivalence methods coming from Pin(2)- equivariant Seiberg-Witten Floer spectra and involutive Heegaard Floer homology.
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