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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 vector field on X × I whose critical points occur only on the boundary. The Morse differential defines a chain map from the Morse complexes 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.

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rdfs:label	Continuation map (en)
rdfs:comment	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 vector field on X × I whose critical points occur only on the boundary. The Morse differential defines a chain map from the Morse complexes 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. (en)
dcterms:subject	Homology theory Morse theory
Wikipage page ID	3313777 (xsd:integer)
Wikipage revision ID	1082537389 (xsd:integer)
Link from a Wikipage to another Wikipage	Vector field Critical point (mathematics) Boundary (topology) Andreas Floer Homology theory Differential topology Floer homology Isomorphism Closed interval Morse theory Chain complex Singular homology Smooth manifold Morse homology Chain homotopy Contact structure Legendrian knots dbr:Morse-Smale
Link from a Wikipage to an external page	https://arxiv.org/abs/math/0308115 https://arxiv.org/abs/math/0407347 https://arxiv.org/abs/math/0407531 https://web.archive.org/web/20050516144821/http:/math.berkeley.edu/%7Ehutching/teach/276/mfp.ps
sameAs	Continuation map Continuation map Continuation map
dbp:wikiPageUsesTemplate	dbt:No_footnotes dbt:Topology-stub
has abstract	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 vector field on X × I whose critical points occur only on the boundary. The Morse differential defines a chain map from the Morse complexes 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). (en)
prov:wasDerivedFrom	wikipedia-en:Continuation_map?oldid=1082537389&ns=0
page length (characters) of wiki page	2252 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Continuation_map
is Link from a Wikipage to another Wikipage of	Morse homology
is foaf:primaryTopic of	wikipedia-en:Continuation_map

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