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2020 Axiomatic $S^1$ Morse–Bott theory
Michael Hutchings, Jo Nelson
Algebr. Geom. Topol. 20(4): 1641-1690 (2020). DOI: 10.2140/agt.2020.20.1641

Abstract

In various situations in Floer theory, one extracts homological invariants from “Morse–Bott” data in which the “critical set” is a union of manifolds, and the moduli spaces of “flow lines” have evaluation maps taking values in the critical set. This requires a mix of analytic arguments (establishing properties of the moduli spaces and evaluation maps) and formal arguments (defining or computing invariants from the analytic data). The goal of this paper is to isolate the formal arguments, in the case when the critical set is a union of circles. Namely, we state axioms for moduli spaces and evaluation maps (encoding a minimal amount of analytical information that one needs to verify in any given Floer-theoretic situation), and using these axioms we define homological invariants. More precisely, we define an (almost) category of “Morse–Bott systems”. We construct a “cascade homology” functor on this category, based on ideas of Bourgeois and Frauenfelder, which is “homotopy invariant”. This machinery is used in our work on cylindrical contact homology.

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Michael Hutchings. Jo Nelson. "Axiomatic $S^1$ Morse–Bott theory." Algebr. Geom. Topol. 20 (4) 1641 - 1690, 2020. https://doi.org/10.2140/agt.2020.20.1641

Information

Received: 27 February 2018; Revised: 12 June 2019; Accepted: 20 August 2019; Published: 2020
First available in Project Euclid: 1 August 2020

zbMATH: 07226702
MathSciNet: MR4127081
Digital Object Identifier: 10.2140/agt.2020.20.1641

Subjects:
Primary: 53D40, 57R58

Rights: Copyright © 2020 Mathematical Sciences Publishers

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Vol.20 • No. 4 • 2020
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