Continuation map
Encyclopedia
In differential topology
, given a family of Morse-Smale functions
on a smooth manifold X parameterized by a closed interval I, one can construct a Morse-Smale vector field
on X × I whose critical points occur only on the boundary. The Morse differential defines a chain map from the Morse complex
es at the boundaries of the family, the continuation map. This can be shown to descend to an isomorphism on Morse homology
, proving its invariance of Morse homology
of a smooth manifold.
Continuation maps were defined by Andreas Floer
to prove the invariance of Floer homology
in infinite dimensional analogues of the situation described above; in the case of finite-dimensional Morse theory, invariance may be proved by proving that Morse homology is isomorphic to singular homology, which is known to be invariant. However, Floer homology is not always isomorphic to a familiar invariant, so continuation maps yield an a priori proof of invariance.
In finite-dimensional Morse theory, different choices made in constructing the vector field on X × I yield distinct but chain homotopic maps and thus descend to the same isomorphism on homology. However, in certain infinite dimensional cases, this does not hold, and these techniques may be used to produce invariants of one-parameter families of objects (such as contact structures or Legendrian knots).
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, given a family of Morse-Smale functions
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...
on a smooth manifold X parameterized by a closed interval I, one can construct a Morse-Smale vector field
Vector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
on X × I whose critical points occur only on the boundary. The Morse differential defines a chain map from the Morse complex
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...
es at the boundaries of the family, the continuation map. This can be shown to descend to an isomorphism on 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...
, proving its invariance 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...
of a smooth manifold.
Continuation maps were defined by Andreas Floer
Andreas Floer
Andreas Floer was a German mathematician who made seminal contributions to the areas of geometry, topology, and mathematical physics, in particular the invention of Floer homology.-Life:...
to prove the invariance of Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
in infinite dimensional analogues of the situation described above; in the case of finite-dimensional Morse theory, invariance may be proved by proving that Morse homology is isomorphic to singular homology, which is known to be invariant. However, Floer homology is not always isomorphic to a familiar invariant, so continuation maps yield an a priori proof of invariance.
In finite-dimensional Morse theory, different choices made in constructing the vector field on X × I yield distinct but chain homotopic maps and thus descend to the same isomorphism on homology. However, in certain infinite dimensional cases, this does not hold, and these techniques may be used to produce invariants of one-parameter families of objects (such as contact structures or Legendrian knots).