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 singular homology. Morse homology also serves as a model for the various infinite-dimensional generalizations known as Floer homology theories.
mathematics, specifically in the field of differential topology, Morsehomology is a homology theory defined for any smooth manifold. It is constructed using...
their homology. Before Morse, Arthur Cayley and James Clerk Maxwell had developed some of the ideas of Morse theory in the context of topography. Morse originally...
finite-dimensional Morsehomology. Andreas Floer introduced the first version of Floer homology, now called symplectic Floer homology, in his proof of the...
result of discrete Morse theory establishes that the CW complex X {\displaystyle {\mathcal {X}}} is isomorphic on the level of homology to a new complex...
See homology for an introduction to the notation. Persistent homology is a method for computing topological features of a space at different spatial resolutions...
proved by proving that Morsehomology is isomorphic to singular homology, which is known to be invariant. However, Floer homology is not always isomorphic...
the A ∞ {\displaystyle A_{\infty }} language first in the context of Morsehomology, and exist in a number of variants. As Fukaya categories are A∞-categories...
Augustin (2004), "Corollary 5.9 (The Preimage Theorem)", Lectures on MorseHomology, Texts in the Mathematical Sciences, vol. 29, Springer, p. 130, ISBN 9781402026959...
Poincaré in 1895 to define homology purely in terms of manifolds (Dieudonné 1989, p. 289). Poincaré simultaneously defined both homology and cobordism, which...
of f of index p (called the Morse number). It computes the (integral) homology of M {\displaystyle M} (cf. Morsehomology): H ∗ ( C ∗ ( f ) ) ≅ H ∗ (...
the persistent homology has emerged through the work of Sergey Barannikov on Morse theory. The set of critical values of smooth Morse function was canonically...
ISBN 0-7923-4475-8. Augustin Banyaga; David Hurtubise (2004). Lectures on MorseHomology. Kluwer Academic Publishers. ISBN 1-4020-2695-1. Augustin Banyaga at...
In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is...
which is analogous to the Morse inequality. This so-called Arnold conjecture triggered the invention of Hamiltonian Floer homology by Andreas Floer in the...
they are all finite. The nth Betti number represents the rank of the nth homology group, denoted Hn, which tells us the maximum number of cuts that can be...
In persistent homology, a persistent homology group is a multiscale analog of a homology group that captures information about the evolution of topological...
topology, including work on involutive Heegaard Floer homology and equivariant Floer homology. She is an associate professor of mathematics at Rutgers...
theory Graph theory Grothendieck's Galois theory Group theory Hodge theory Homology theory Homotopy theory Ideal theory Index theory Information theory Invariant...
ISBN 978-0-691-15423-7. Berg, Michael (2 February 2014). "MAA Book Review: Morse Theory and Floer Homology". Retrieved 11 December 2021. O'Connor, John J.; Robertson...
the homology, cohomology, and homotopy groups of X determine those of Y. A result of this kind was first stated by Solomon Lefschetz for homology groups...
structure on the sphere. In light of the smooth structure, the existence of a Morse function would show RPn is a CW complex. One such function is given by,...
circle, S 1 {\displaystyle S^{1}} . Therefore the fundamental group and homology groups are isomorphic to those of the circle: π 1 ( S 1 × D 2 ) ≅ π 1 (...
the effects on the homology, homotopy groups, or other invariants of the manifold are known. A relatively easy argument using Morse theory shows that a...
solutions of holonomic D-modules. A key observation was that the intersection homology of Mark Goresky and Robert MacPherson could be described using sheaf complexes...
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