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Monotone Lagrangians in ${\mathbb{C}}{\mathbb{P}}^n$ of minimal Maslov number n + 1

Part of: Symplectic geometry, contact geometry

Published online by Cambridge University Press:  21 February 2020

MOMCHIL KONSTANTINOV  and
JACK SMITH
MOMCHIL KONSTANTINOV
Affiliation:
Department of Mathematics, University College London, Gower Street, London, WC1E 6BT. e-mail: momchil.konstantinov.14@ucl.ac.uk
JACK SMITH
Affiliation:
St John’s College, Cambridge, CB2 1TP. e-mail: j.smith@dpmms.cam.ac.uk
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Abstract

We show that a monotone Lagrangian L in ${\mathbb{C}}{\mathbb{P}}^n$ of minimal Maslov number n + 1 is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to ${\mathbb{R}}{\mathbb{P}}^n$. To prove this we use Zapolsky’s canonical pearl complex for L over ${\mathbb{Z}}$, and twisted versions thereof, where the twisting is determined by connected covers of L. The main tool is the action of the quantum cohomology of ${\mathbb{C}}{\mathbb{P}}^n$ on the resulting Floer homologies.

MSC classification

Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 53D40: Floer homology and cohomology, symplectic aspects
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 1 , July 2021 , pp. 1 - 21
Copyright
© Cambridge Philosophical Society 2020

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