#### Geometry & Topology, Vol. 6 (2002)
Paper no. 7, pages 195--218.

## Homotopy type of symplectomorphism groups of S^2 X S^2

### Silvia Anjos

**Abstract**.
In this paper we discuss the topology of the symplectomorphism group
of a product of two 2-dimensional spheres when the ratio of their
areas lies in the interval (1,2]. More precisely we compute the
homotopy type of this symplectomorphism group and we also show that
the group contains two finite dimensional Lie groups generating the
homotopy. A key step in this work is to calculate the mod 2 homology
of the group of symplectomorphisms. Although this homology has a
finite number of generators with respect to the Pontryagin product, it
is unexpected large containing in particular a free noncommutative
ring with 3 generators.
**Keywords**.
Symplectomorphism group, Pontryagin ring, homotopy equivalence

**AMS subject classification**.
Primary: 57S05, 57R17.
Secondary: 57T20, 57T25.

**DOI:** 10.2140/gt.2002.6.195

**E-print:** `arXiv:math.SG/0009220`

Submitted to GT on 1 October 2001.
(Revised 11 March 2002.)
Paper accepted 26 April 2002.
Paper published 27 April 2002.

Notes on file formats
Silvia Anjos

Departamento de Matematica

Instituto Superior Tecnico, Lisbon, Portugal

Email: sanjos@math.ist.utl.pt

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