Floer theory and topology of $\mathrm{Diff} (S^2)$

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 $i : \mathrm{Diff}^1 (S^2) \to \mathrm{Diff} (S^2)$ vanishes on all homotopy groups, where $\mathrm{Diff}^1 (S^2) \subset \mathrm{Diff}(S^2)$ 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.

Keywords

Floer theory, positivity of intersections, groups of diffeomorphisms

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Received 20 August 2015

Published 8 September 2017