A construction of the quantum Steenrod squares and their algebraic relations

Wilkins, Nicholas

doi:10.2140/gt.2020.24.885

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A construction of the quantum Steenrod squares and their algebraic relations

Nicholas Wilkins

Geometry & Topology 24 (2020) 885–970

DOI: 10.2140/gt.2020.24.885

Abstract

We construct a quantum deformation of the Steenrod square construction on closed monotone symplectic manifolds, based on the work of Fukaya, Betz and Cohen. We prove quantum versions of the Cartan and Adem relations. We compute the quantum Steenrod squares for all ${ℂ ℙ}^{n}$ and give the means of computation for all toric varieties. As an application, we also describe two examples of blowups along a subvariety, in which a quantum correction of the Steenrod square on the blowup is determined by the classical Steenrod square on the subvariety.

Keywords

Gromov–Witten theory, quantum cohomology, Steenrod squares, symplectic geometry, symplectic topology

Mathematical Subject Classification 2010

Primary: 53D45

Secondary: 14N35, 55S10

References

Publication

Received: 15 May 2018

Revised: 1 September 2019

Accepted: 2 October 2019

Published: 23 September 2020

Proposed: Leonid Polterovich

Seconded: Ralph Cohen, Ciprian Manolescu

Authors

Nicholas Wilkins
School of Mathematics University of Bristol Bristol United Kingdom

Volume 24, issue 2 (2020)