Circle actions and Suspension operations on Smooth manifolds

TL;DR

This paper studies suspension operations on smooth manifolds with circle actions, showing how surgeries along circles produce new manifolds and aid in their classification.

Contribution

It introduces and analyzes suspension operations on manifolds, demonstrating their role in constructing and classifying manifolds with free circle actions.

Findings

Surgeries along circles produce manifolds called suspensions.

Suspension operations are fundamental in classifying manifolds with free S^1-actions.

Applications illustrate the usefulness of suspension in manifold theory.

Abstract

Let $M$ be a smooth manifold with $\dim M\geq 3$ and a base point $x_{0}$. Surgeries along the oriented circle $S^{1}\times \{x_{0}\}$ on the product $ S^{1}\times M$ yields two manifolds $\Sigma _{0}M$ and $\Sigma _{1}M$, called the suspensions of $M$. The suspension operations $\Sigma _{i}$ play a basic role in the construction and classification of the smooth manifolds which admit free $ S^{1}$-actions. We illustrate this by a number of applications.