Morse theory | Homology theory
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). (Wikipedia).
Determining Supplementary Angles from Parallel Lines and a Transversal
👉 Learn how to identify angles from a figure. This video explains how to solve problems using angle relationships between parallel lines and transversal. We'll determine the solution given, corresponding, alternate interior and exterior. All the angle formed by a transversal with two paral
From playlist Parallel Lines and a Transversal
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From playlist Parallel Lines and a Transversal
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From playlist Parallel Lines and a Transversal
Geometry - Identifying Corresponding Angles from a Figure
👉 Learn how to identify angles from a figure. This video explains how to solve problems using angle relationships between parallel lines and transversal. We'll determine the solution given, corresponding, alternate interior and exterior. All the angle formed by a transversal with two paral
From playlist Parallel Lines and a Transversal
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From playlist Parallel Lines and a Transversal
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From playlist Parallel Lines and a Transversal
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From playlist Parallel Lines and a Transversal
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From playlist MAST30026 Metric and Hilbert spaces
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From playlist What is a Manifold?
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From playlist MAST30026 Metric and Hilbert spaces
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From playlist Parallel Lines and a Transversal
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From playlist Parallel Lines and a Transversal
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From playlist Algebraic Topology
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From playlist MAST30026 Metric and Hilbert spaces
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From playlist Manifolds
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From playlist 2017 - T3 - Analysis in Quantum Information Theory - CEB Trimester
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From playlist Parallel Lines and a Transversal
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From playlist Manifolds