Betti number

Betti numbers are geometric invariants widely used in various branches of mathematics, including geometry, algebra, algebraic geometry, arithmetic geometry, and topological data analysis.Betti Number in Wolfram Mth World. It is originally defined for a topological space as the ranks of the Betti s, i.e. the singular homology groups, but the construction also works for (discrete) groups \(G\), thanks to the study of classification space \(BG\) and the Eilenberg-MacLane space \(K(G,1)\). The reader should be careful that the Betti homology of a discrete group is not defined as the Betti homology of the underlying topological space, which is trivial at positive degree.
== Applications in googology ==
Using betti numbers \((b_k(G))_{k \in \mathbb{N}}\) for finitely presented groups \(G\), Alexander Nabutovsky and Shmuel Weinberger constructed a system \((b_k(N))_{k \in \mathbb{N}}\) of functions including uncomputable fast-growing functions.Alexander Nabutovsky and Shmuel Weinberger, Betti numbers of finitely presented groups and very rapidly growing functions, 
Topology, Volume 46, Issue 2, March 2007, pp 211--223., in response to a challenge by Scott Aaronson to find functions that outgrow the busy beaver function arising in popular areas of math.P. Michel, The Busy Beaver Competition: A historical survey (pp 2--3). arXiv, accessed 2022-06-24.
Let \(H_k(G)\) be the \(k\)-th homology group of a group \(G\) in the sense explained above. Given a finite \(G\) (with a fixed presentation), we define its length as the sum of the lengths of all its relators plus the number of generators. The \(k\)-th Betti number of a finitely presented group \(G\) is defined as \(b_k(G) = \text{rank }H_k(G)\) (using the ). Given a nonnegative integer \(N\), we define the number \(b_k(N)\) as the maximum of finite Betti numbers \(b_k(G)\) where \(G\) runs through finitely presented groups with length at most \(N\).
Specifically, a relation between \(b_k\) for \(k\geq 3\) and the Busy Beaver function for Turing machines with an oracle for the halting problem (here called "\(\Sigma_3\)") was given in a preprint version of the original paper. This relation was consists of showing that there are computable \(f,g\) such that for all natural \(n\), \(b_k\circ g(n)\) upper-bounds \(\Sigma_3(n)\), while \(\Sigma_3\circ f(n)\) upper-bounds \(b_k(n)\). In the authors' words, for each natural \(k\geq 3\) we have that each \(b_k\) "grows as the third-order Busy Beaver function."A. Nabutovsky, S. Weinberger, Betti numbers of finitely presented groups and very rapidly growing functions (pages 8, 12), arXiv version, submitted 2006. Accessed 2022-06-24.
== Sources ==
== See also ==
Category:Functions
Category:Uncomputable