Journal of Symplectic Geometry

Volume 15 (2017)

Number 3

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

Pages: 853 – 859



Yasha Savelyev (CUICBAS, University of Colima, Mexico)


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.


Floer theory, positivity of intersections, groups of diffeomorphisms

Received 20 August 2015

Published 8 September 2017