Volume 24, issue 2 (2020)

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A construction of the quantum Steenrod squares and their algebraic relations

Nicholas Wilkins

Geometry & Topology 24 (2020) 885–970

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 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.

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Gromov–Witten theory, quantum cohomology, Steenrod squares, symplectic geometry, symplectic topology
Mathematical Subject Classification 2010
Primary: 53D45
Secondary: 14N35, 55S10
Received: 15 May 2018
Revised: 1 September 2019
Accepted: 2 October 2019
Published: 23 September 2020
Proposed: Leonid Polterovich
Seconded: Ralph Cohen, Ciprian Manolescu
Nicholas Wilkins
School of Mathematics
University of Bristol
United Kingdom