Outdated Archival Version

These pages are not updated anymore. They reflect the state of 6 June 2010. For the current production of this journal, please refer to http://www.springer.com/mathematics/geometry/journal/40062.

On the Betti numbers of a loop space

Samson Saneblidze

Let $A$ be a special homotopy G-algebra over a commutative unital ring $\Bbbk$ suchthat both $H(A)$ and $\operatorname{Tor}_{i}^{A}(\Bbbk,\Bbbk)$ are finitely generated$\Bbbk$-modules for all $i$, and let $\tau_{i}(A)$ be the cardinality of a minimalgenerating set for the $\Bbbk$-module $\operatorname{Tor}_{i}^{A}(\Bbbk,\Bbbk).$ Thenthe set $\left\{ \tau _{i}(A)\right\} $ is unbounded if and only if $\tilde{H}(A)$has two or more algebra generators. When $A=C^{\ast}(X;\Bbbk)$ is the simplicialcochain complex of a simply connected finite $CW$-complex $X,$ there is a similarstatement for the "Betti numbers" of the loop space $\Omega X.$ This unifies existingproofs over a field $\Bbbk$ of zero or positive characteristic.

Journal of Homotopy and Related Structures, Vol. 5(2010), No. 1, pp. 1-13