# Fixed point sets of isotopies on surfaces

### François Béguin

Université Paris 13, Villetaneuse, France### Sylvain Crovisier

Université Paris Sud, Orsay, France### Frédéric Le Roux

Université Pierre et Marie Curie, Paris, France

## Abstract

We consider a self-homeomorphism $h$ of some surface $S$. A subset $F$ of the fixed point set of $h$ is said to be *unlinked* if there is an isotopy from the identity to $h$ that fixes every point of $F$. With Le Calvez’ transverse foliations theory in mind, we prove the existence of unlinked sets that are maximal with respect to inclusion. As a byproduct, we prove the arcwise connectedness of the space of homeomorphisms of the 2-sphere that preserve orientation and pointwise fix some given closed connected set $F$.

## Cite this article

François Béguin, Sylvain Crovisier, Frédéric Le Roux, Fixed point sets of isotopies on surfaces. J. Eur. Math. Soc. 22 (2020), no. 6, pp. 1971–2046

DOI 10.4171/JEMS/960