Volume 20, issue 4 (2020)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Axiomatic $S^1$ Morse–Bott theory

Michael Hutchings and Jo Nelson

Algebraic & Geometric Topology 20 (2020) 1641–1690
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.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/agt

We have not been able to recognize your IP address 3.236.222.124 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords
Morse–Bott theory, cascade homology, contact homology
Mathematical Subject Classification 2010
Primary: 53D40, 57R58
Publication
Received: 27 February 2018
Revised: 12 June 2019
Accepted: 20 August 2019
Published: 20 July 2020
Authors
 Michael Hutchings Department of Mathematics University of California, Berkeley Berkeley, CA United States Jo Nelson Department of Mathematics Rice University Houston, TX United States