= First Betti-number bound from polynomial fundamental-group growth
{title2=$b_1(M)\leq\dim M$}
For a compact connected <Riemannian manifold> with nonnegative <Ricci curvature>, its finitely generated <fundamental group> acts freely and <properly discontinuously> on the complete <universal cover>. The preceding <polynomial growth of properly discontinuous isometry groups> bound applies. The degree-one <Hurewicz theorem> identifies first integral <homology> with the <abelianization> of the fundamental group. If its free rank is $b_1(M)$, this group maps onto $\mathbb Z^{b_1(M)}$. Choosing lifts of a basis gives at least a constant times $m^{b_1(M)}$ distinct elements in a word ball of radius a constant times $m$. The upper bound of order $m^{\dim M}$ forces the displayed inequality.
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