circle-valued Morse theory

In mathematics, circle-valued Morse theory studies the topology of a smooth manifold by analyzing the critical points of smooth maps from the manifold to the circle, in the framework of Morse homology.{{citation

| last = Pajitnov | first = Andrei V.

| doi = 10.1515/9783110197976

| isbn = 978-3-11-015807-6

| mr = 2319639

| publisher = Walter de Gruyter & Co., Berlin

| series = de Gruyter Studies in Mathematics

| title = Circle-valued Morse theory

| volume = 32

| year = 2006}}. It is an important special case of Sergei Novikov's Morse theory of closed one-forms.{{citation

| last = Farber | first = Michael

| doi = 10.1090/surv/108

| isbn = 0-8218-3531-9

| mr = 2034601

| page = 50

| publisher = American Mathematical Society, Providence, RI

| series = Mathematical Surveys and Monographs

| title = Topology of closed one-forms

| url = https://books.google.com/books?id=unm2pS8WiNsC&pg=PA50

| volume = 108

| year = 2004| doi-access = free

}}.

Michael Hutchings and Yi-Jen Lee have connected it to Reidemeister torsion and Seiberg–Witten theory.{{citation

| last1 = Hutchings | first1 = Michael | authorlink = Michael Hutchings (mathematician)

| last2 = Lee | first2 = Yi-Jen

| doi = 10.1016/S0040-9383(98)00044-5

| issue = 4

| journal = Topology

| mr = 1679802

| pages = 861–888

| title = Circle-valued Morse theory, Reidemeister torsion, and Seiberg-Witten invariants of 3-manifolds

| volume = 38

| year = 1999| arxiv = dg-ga/9612004

| s2cid = 12740033 }}.