mathematics, specifically in the field of differential topology, Morse homology 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 Morse homology. Andreas Floer introduced the first version of Floer homology, now called symplectic Floer homology, in his proof of the...

Borel–Moore homology Cellular homology Cyclic homology Hochschild homology Floer homology Intersection homology K-homology Khovanov homology Morse homology Persistent...

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 Morse homology is isomorphic to singular homology, which is known to be invariant. However, Floer homology is not always isomorphic...

manifold to the circle, in the framework of Morse homology. It is an important special case of Sergei Novikov's Morse theory of closed one-forms. Michael Hutchings...

Augustin (2004), "Corollary 5.9 (The Preimage Theorem)", Lectures on Morse Homology, Texts in the Mathematical Sciences, vol. 29, Springer, p. 130, ISBN 9781402026959...

the A ∞ {\displaystyle A_{\infty }} language first in the context of Morse homology, and exist in a number of variants. As Fukaya categories are A∞-categories...

of f of index p (called the Morse number). It computes the (integral) homology of M {\displaystyle M} (cf. Morse homology): H ∗ ( C ∗ ( f ) ) ≅ H ∗ (...

Poincaré in 1895 to define homology purely in terms of manifolds (Dieudonné 1989, p. 289). Poincaré simultaneously defined both homology and cobordism, which...

introduced a type of homology on a 3-manifolds defined in analogy with Morse homology in finite dimensions. In this homology theory, the Morse function is the...

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...

In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is...

362–386. doi:10.1090/s0002-9947-1942-0006479-x. MR 0006479. Morse, M. (1952). "Homology relations on regular orientable manifolds". Proc Natl Acad Sci...

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...

which is analogous to the Morse inequality. This so-called Arnold conjecture triggered the invention of Hamiltonian Floer homology by Andreas Floer in the...

ISBN 0-7923-4475-8. Augustin Banyaga; David Hurtubise (2004). Lectures on Morse Homology. Kluwer Academic Publishers. ISBN 1-4020-2695-1. Augustin Banyaga at...

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...

topology, including work on involutive Heegaard Floer homology and equivariant Floer homology. She is an associate professor of mathematics at Rutgers...

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...

In persistent homology, a persistent homology group is a multiscale analog of a homology group that captures information about the evolution of topological...

two-chambered soap bubbles, and for his work on circle-valued Morse theory and on embedded contact homology, which he defined. As an undergraduate student at Harvard...

theory Graph theory Grothendieck's Galois theory Group theory Hodge theory Homology theory Homotopy theory Ideal theory Index theory Information theory Invariant...

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...

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 (...

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...

theory- anomaly and obstruction, 2007 Morse homotopy, A ∞ {\displaystyle A_{\infty }} -Category, and Floer homologies, in H. J. Kim (editor) Proceedings...