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On the Betti numbers of a loop space
#
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