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- Continuation_map 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 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 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).".
- Continuation_map wikiPageExternalLink 0407347.
- Continuation_map wikiPageExternalLink 0308115.
- Continuation_map wikiPageExternalLink 0407531.
- Continuation_map wikiPageExternalLink mfp.ps.
- Continuation_map wikiPageID "3313777".
- Continuation_map wikiPageLength "2220".
- Continuation_map wikiPageOutDegree "20".
- Continuation_map wikiPageRevisionID "571974224".
- Continuation_map wikiPageWikiLink Andreas_Floer.
- Continuation_map wikiPageWikiLink Boundary_(topology).
- Continuation_map wikiPageWikiLink Category:Homology_theory.
- Continuation_map wikiPageWikiLink Category:Morse_theory.
- Continuation_map wikiPageWikiLink Chain_complex.
- Continuation_map wikiPageWikiLink Chain_homotopy.
- Continuation_map wikiPageWikiLink Closed_interval.
- Continuation_map wikiPageWikiLink Contact_geometry.
- Continuation_map wikiPageWikiLink Contact_structure.
- Continuation_map wikiPageWikiLink Critical_point_(mathematics).
- Continuation_map wikiPageWikiLink Differentiable_manifold.
- Continuation_map wikiPageWikiLink Differential_topology.
- Continuation_map wikiPageWikiLink Floer_homology.
- Continuation_map wikiPageWikiLink Homotopy_category_of_chain_complexes.
- Continuation_map wikiPageWikiLink Interval_(mathematics).
- Continuation_map wikiPageWikiLink Isomorphism.
- Continuation_map wikiPageWikiLink Legendrian_knot.
- Continuation_map wikiPageWikiLink Legendrian_knots.
- Continuation_map wikiPageWikiLink Morse-Smale.
- Continuation_map wikiPageWikiLink Morse_homology.
- Continuation_map wikiPageWikiLink Singular_homology.
- Continuation_map wikiPageWikiLink Smooth_manifold.
- Continuation_map wikiPageWikiLink Vector_field.
- Continuation_map wikiPageWikiLinkText "continuation map".
- Continuation_map hasPhotoCollection Continuation_map.
- Continuation_map wikiPageUsesTemplate Template:Topology-stub.
- Continuation_map subject Category:Homology_theory.
- Continuation_map subject Category:Morse_theory.
- Continuation_map 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 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 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_map label "Continuation map".
- Continuation_map sameAs m.0953d9.
- Continuation_map sameAs Q5165380.
- Continuation_map sameAs Q5165380.
- Continuation_map wasDerivedFrom Continuation_map?oldid=571974224.
- Continuation_map isPrimaryTopicOf Continuation_map.