Roughly, the number of k-dimensional holes on a topological surface
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and 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 made before separating a surface into two pieces or 0-cycles, 1-cycles, etc.[1] For example, if then , if then , if then , if then , etc. Note that only the ranks of infinite groups are considered, so for example if , where is the finite cyclic group of order 2, then . These finite components of the homology groups are their torsion subgroups, and they are denoted by torsion coefficients.
The term "Betti numbers" was coined by Henri Poincaré after Enrico Betti. The modern formulation is due to Emmy Noether. Betti numbers are used today in fields such as simplicial homology, computer science and digital images.
^Barile, and Weisstein, Margherita and Eric. "Betti number". From MathWorld--A Wolfram Web Resource.
of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite. The nth Bettinumber represents...
In persistent homology, a persistent Bettinumber is a multiscale analog of a Bettinumber that tracks the number of topological features that persist...
Betti may refer to: Betti (given name) Betti (surname) Bettinumber in topology, named for Enrico BettiBetti's theorem in engineering theory, named for...
neuroscience, topological quantities like the Euler characteristic and Bettinumber have been used to measure the complexity of patterns of activity in neural...
complexity of the program is equal to the cyclomatic number of its graph (also known as the first Bettinumber), which is defined as M = E − N + P . {\displaystyle...
this complex is intrinsic to R, one may define the graded Betti numbers βi, j as the number of grade-j images coming from Fi (more precisely, by thinking...
others are linear combinations. In particular, this implies that the 1st Bettinumber of a 2-torus is two. More generally, on an n {\displaystyle n} -dimensional...
diffeomorphic to Rn if it has positive curvature at only one point. Gromov's Bettinumber theorem. There is a constant C = C(n) such that if M is a compact connected...
other i. Therefore X is a connected space, with one non-zero higher Bettinumber, namely, b n = 1 {\displaystyle b_{n}=1} . It does not follow that X...
connection, the cyclomatic number of a graph G is also called the first Bettinumber of G. More generally, the first Bettinumber of any topological space...
a complex, specifically the Bettinumber and torsion coefficients of a dimension of the complex, where the Bettinumber corresponds to the rank of the...
H 1 ( X , Z ) = 0 {\displaystyle H^{1}(X,\mathbb {Z} )=0} . Thus the Bettinumber b 1 ( X ) {\displaystyle b_{1}(X)} is zero, and by Poincaré duality,...
Laura Betti (née Trombetti; 1 May 1927 – 31 July 2004) was an Italian actress known particularly for her work with directors Federico Fellini, Pier Paolo...
1967 when the rest of Sweden had left-hand driving. These roads are mostly number construction[clarification needed] and do not have special privileges. Road...
supermultiplet, a number of chiral supermultiplets equal to the third Bettinumber of the G 2 {\displaystyle G_{2}} manifold and a number of U(1) vector...
the torsion subgroup of an abelian group is a torsion abelian group. Bettinumber Dummit, David; Foote, Richard. Abstract Algebra, ISBN 978-0471433347...
Bettinumber: Hopf surfaces Inoue surfaces; several other families discovered by Inoue have also been called "Inoue surfaces" Positive second Betti number:...
The nonnegative integer r {\displaystyle r} is called the free rank or Bettinumber of the module M {\displaystyle M} , while a 1 , … , a m {\displaystyle...
that, for a closed, oriented manifold, the Betti numbers derived through simplicial homology were the same Betti numbers as those derived through de Rham...
multiplication by exp(n). The quotient is a complex manifold whose first Bettinumber is one, so by the Hodge theory, it cannot be Kähler. A Calabi–Yau manifold...
3-folds with second Bettinumber 1 into 17 classes, and Mori & Mukai (1981) classified the smooth ones with second Bettinumber at least 2, finding 88...
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