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.

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Silvia Anjos
Departamento de Matematica
Instituto Superior Tecnico, Lisbon, Portugal

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