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