Lower bounds for the complexity of three-dimensional manifolds. (Q1432545)

The first author introduced the complexity \(c(M)\) of a manifold \(M\) in [Acta Appl. Math. 19, 101--130 (1990; Zbl 0724.57012)]. While it is relatively easy to find upper bounds for the complexity, no lower bounds are known yet. In the paper under review, the authors obtain the following two results: Theorem 1: Let \(M\) be a closed irreducible orientable three-manifold different from \(S^3\), \(\mathbb{R} P^3\) and \(L_{3, 1}\). Then \(c(M)\geq 2\log_5| \text{Tor} (H_1(M)) |+ \beta\) where \(\beta\) denotes the first Betti Number of \(M\). Theorem 2: Under the conditions of Theorem 1, it holds that \(c(M)\geq -1+\frac{c(\pi_1(M))} {3}\) where the complexity of a group \(G\) is the minimum of the length of all of its finite presentations.