In
differential topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which ...
, given a family of
Morse-Smale functions on a
smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may ...
''X'' parameterized by a
closed interval
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
''I'', one can construct a
Morse-Smale vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...
on ''X'' × ''I'' whose
critical points occur only on the
boundary. The Morse differential defines a
chain map
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
from the
Morse complexes at the boundaries of the family, the continuation map. This can be shown to descend to an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
on
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 t ...
, proving its invariance of Morse homology of a smooth manifold.
Continuation maps were defined by
Andreas Floer
Andreas Floer (; 23 August 1956 – 15 May 1991) was a German mathematician who made seminal contributions to symplectic topology, and mathematical physics, in particular the invention of Floer homology. Floer's first pivotal contribution was a s ...
to prove the invariance of
Floer homology
In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is an invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer intro ...
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
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension n, the n-dimensional ...
, 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 structure
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution ...
s or
Legendrian knots).
References
Lecture Notes on Morse Homology (including continuation maps in finite-dimensional theory), by Michael HutchingsContact homology and homotopy groups of the space of contact structures by Frederic BourgeoisContact homology and one parameter families of Legendrian knots by Tamas KalmanFloer homology of families I, by Michael Hutchings
Morse theory
Homology theory
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