Finite type invariants of integral homology 3-spheres: A survey

TL;DR

This survey reviews the state of finite type invariants for integral homology 3-spheres and introduces a new result showing their algebraic structure as a graded polynomial algebra generated by additive invariants.

Contribution

It presents a new proof that the space of finite type invariants forms a graded polynomial algebra generated by additive invariants.

Findings

The space of finite type invariants is a graded polynomial algebra.

Additive invariants generate the algebra of finite type invariants.

Open questions remain in the study of these invariants.

Abstract

This is a survey on the current status of the study of finite type invariants of integral homology 3-spheres based on lectures given in the workshop on knot theory at Banach International Center of Mathematics, Warsaw, July 1995. As a new result, we show that the space of finite type invariants of integral homology 3-spheres is a graded polynomial algebra generated by invariants additive under the connected sum. We also discuss some open questions on this subject.