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arXiv · 2008.10412

Infinitely many non-isotopic real symplectic forms on $S^2 \times S^2$

Gleb Smirnov
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Abstract

Let $(S^2,ω)$ be a symplectic sphere, and let $τ\colon S^2 \to S^2$ be an anti-symplectic involution of $(S^2,ω)$. We consider the product $(S^2,ω) \times (S^2,ω)$ endowed with the anti-symplectic involution $τ\times τ$, and study the space of monotone anti-invariant symplectic forms on this four-manifold. We show that this space is disconnected. In addition, during the course of the proof, we produce a diffeomorphism of the grassmannian (2,4) which induces the identity map on all homology and homotopy groups, but which is not homotopic to the identity.

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Gleb Smirnov. 2020-09-11. Infinitely many non-isotopic real symplectic forms on $S^2 \times S^2$. https://doi.org/10.1112/topo.12176

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