On the Betti numbers of a loop space

Samson Saneblidze

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arXiv · 0905.2591

On the Betti numbers of a loop space

Samson Saneblidze

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Abstract

Let $A$ be a special homotopy G-algebra over a commutative unital ring $\Bbbk$ such that both $H(A)$ and $\operatorname{Tor}_{i}^{A}(\Bbbk,\Bbbk)$ are finitely generated $\Bbbk$-modules for all $i$, and let $τ_{i}(A)$ be the cardinality of a minimal generating set for the $\Bbbk$-module $\operatorname{Tor}_{i}^{A}(\Bbbk,\Bbbk).$ Then the set ${τ_{i}(A)} $ is unbounded if and only if $\tilde{H}(A)$ has two or more algebra generators. When $A=C^{\ast}(X;\Bbbk)$ is the simplicial cochain complex of a simply connected finite $CW$-complex $X,$ there is a similar statement for the "Betti numbers" of the loop space $ΩX.$ This unifies existing proofs over a field $\Bbbk$ of zero or positive characteristic.

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Samson Saneblidze. 2009-12-24. On the Betti numbers of a loop space. https://arxiv.org/abs/0905.2591

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