Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Betti numbers of a loop space   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.

11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
24
The categories of this paper are: dg-ga
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
33
The categories of this paper are: plasm-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
13
The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
14
The categories of this paper are: math-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: eess