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Title: Quantum Steenrod squares, related operations, and their properties
Author: Wilkins, Nicholas
ISNI:       0000 0004 8502 6639
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
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In this thesis, we generalise the Steenrod square on the cohomology of a topological space to a quantum Steenrod square on the quantum cohomology of a symplectic manifold. We prove quantum versions of the Cartan and Adem relations, and use the former to calculate the quantum Steenrod square for CPn, as well as more generally giving the means to compute the operation for all monotone toric varieties. We calculate the quantum Steenrod squares for 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. We then relate the quantum Steenrod square to Seidel's equivariant pair-of-pants product for open convex weakly monotone symplectic manifolds, using an equivariant version of the PSS isomorphism. We proceed similarly for Z/2-equivariant symplectic cohomology, using an equivariant version of the c*-map. We prove a symplectic Cartan relation, pointing out the difficulties in stating it. We will give some chain level calculations of the equivariant pair-of-pants product for T*Sn. We calculate the quantum Steenrod square for the negative line bundles O(-1) → CPm, and use this to calculate the equivariant pair-of-pants square by extending previous results due to Ritter. We finish with a brief sketch of further ideas, involving a Z/2-equivariant version of the Chas-Sullivan product and the Viterbo isomorphism.
Supervisor: Ritter, Alexander Sponsor: Engineering and Physical Sciences Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Symplectic geometry