page_title: Betti number
section_title: Geometric interpretation
hierarchical_section_title: Betti number / Geometric interpretation
context_page_description: 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, the sequence of Betti numbers is 0 from some point onward, and they are all finite.
  The nᵗʰ Betti number represents the rank of the nᵗʰ homology group, denoted Hₙ, which tells us the maximum amount of cuts that must be made before separating a surface into two pieces or 0-cycles, 1-cycles, etc. 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 Z₂ is the finite cyclic group of order 2, then. These finite components of the homology groups are their torsion subgroups, are 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, digital images, etc.
context_section_description: Informally, the kth Betti number refers to the number of k-dimensional holes on a topological surface. A "k-dimensional hole" is a k-dimensional cycle that is not a boundary of a (k+1)-dimensional object.
  The first few Betti numbers have the following definitions for 0-dimensional, 1-dimensional, and 2-dimensional simplicial complexes:
  b₀ is the number of connected components;
  b₁ is the number of one-dimensional or "circular" holes;
  b₂ is the number of two-dimensional "voids" or "cavities".
  Thus, for example, a torus has one connected surface component so b₀ = 1, two "circular" holes (one equatorial and one meridional) so b₁ = 2, and a single cavity enclosed within the surface so b₂ = 1.
  Another interpretation of bₖ is the maximum number of k-dimensional curves that can be removed while the object remains connected. For example, the torus remains connected after removing two 1-dimensional curves (equatorial and meridional) so b₁ = 2.
  The two-dimensional Betti numbers are easier to understand because we see the world in 0, 1, 2, and 3-dimensions; however, the succeeding Betti numbers are of higher-dimension than apparent physical space.
caption: For a torus, the first Betti number is b1 = 2 , which can be intuitively thought of as the number of circular "holes"
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