Circle-valued Morse theory
Encyclopedia
In mathematics
, circle-valued Morse theory studies the topology of a smooth manifold by analyzing the critical point
s of smooth maps from the manifold to the circle
, in the framework of Morse homology
. It is an important special case of Sergei Novikov's Morse theory
of closed one-form
s.
Michael Hutchings and Yi-Jen Lee have connected it to Reidemeister torsion and Seiberg-Witten theory.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, circle-valued Morse theory studies the topology of a smooth manifold by analyzing the critical point
Critical point (mathematics)
In calculus, a critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is 0. The value of the function at a critical point is a critical value of the function...
s of smooth maps from the manifold to the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
, in the framework of Morse homology
Morse homology
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to...
. It is an important special case of Sergei Novikov's Morse theory
Morse theory
In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
of closed one-form
One-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
s.
Michael Hutchings and Yi-Jen Lee have connected it to Reidemeister torsion and Seiberg-Witten theory.