On the Betti numbers of a loop space

TL;DR

This paper investigates the growth of Betti numbers of loop spaces, establishing a connection between their unboundedness and the number of algebra generators of the cohomology algebra, unifying previous results over various fields.

Contribution

It provides a unified criterion linking Betti number unboundedness to the algebraic generators of cohomology for simply connected finite CW-complexes.

Findings

Betti numbers of loop spaces are unbounded iff cohomology has multiple generators.

Unifies proofs over fields of different characteristics.

Connects algebraic generators with topological invariants.

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 $\tau_{i}(A)$ be the cardinality of a minimal generating set for the $\Bbbk$-module $\operatorname{Tor}_{i}^{A}(\Bbbk,\Bbbk).$ Then the set ${\tau_{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 $\Omega X.$ This unifies existing proofs over a field $\Bbbk$ of zero or positive characteristic.