Floer theory and topology of $Diff (S^2)$

TL;DR

This paper employs pseudo-holomorphic curves to demonstrate that the inclusion of diffeomorphisms with a fixed non-degenerate point into all orientation-preserving diffeomorphisms of the sphere induces trivial maps on all homotopy groups, revealing new topological insights.

Contribution

It proves that the inclusion map from diffeomorphisms with a fixed non-degenerate point into the full diffeomorphism group of the sphere induces trivial homotopy maps, using Floer theory techniques.

Findings

Inclusion map induces zero on all homotopy groups

Uses pseudo-holomorphic curves in Floer theory

Complements classical results of Smale and Eels-Earl

Abstract

We say that a fixed point of a diffeomorphism is non-degenerate if 1 is not an eigenvalue of the linearization at the fixed point. We use pseudo-holomorphic curves techniques to prove the following: the inclusion map $$i: \text{Diff} ^{1} (S ^{2} ) \to \text{Diff} (S^2)$$ vanishes on all homotopy groups, where $\text{Diff} ^{1} (S^{2} ) \subset \text {Diff} (S^{2} )$ denotes the space of orientation preserving diffeomorphisms of $S ^{2} $ with a prescribed non-degenerate fixed point. This complements the classical results of Smale and Eels and Earl.