Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing $$ H^1(X) \times \pi_1(X) \to \mathbb R $$ which is denoted by $\langle,\rangle$. Now, let $\alpha\in H^1(X)$ and $[\gamma]\in\pi_1(X)$. If my memory was right there exists a representative loop $\gamma$ which is smooth and of minimal length. And, the metric $g$ naturally induces a norm $\|\cdot\|_g$ on $H^1(X)$ by taking the infimum of all $\|a\|_{L^\infty;g}$ over all 1-forms $a$ representing $\alpha$. We use $L_g$ to denote the length with respect to $g$. Then my guess is as follows:

Question. Can we find a constant $c$ so that $$ \langle \alpha, [\gamma] \rangle \le c \|\alpha\|_g \cdot L_g(\gamma) $$ for any $\alpha$ and $[\gamma]$? If so, can we simply take $c=1$?

Let $(X,g)$ be a compact smooth Riemannian manifold. It is known that $H^1(X, \mathbb R)\cong \mathrm{Hom} (\pi_1(X), \mathbb R)$, namely there is a natural pairing $$ H^1(X) \times \pi_1(X) \to \mathbb R $$ which is denoted by $\langle,\rangle$. Now, let $\alpha\in H^1(X)$ and $[\gamma]\in\pi_1(X)$. Take a representative loop $\gamma$. And, the metric $g$ naturally induces a norm $\|\cdot\|_g$ on $H^1(X)$ by taking the infimum of all $\|a\|_{L^\infty;g}$ over all 1-forms $a$ representing $\alpha$. We use $L_g$ to denote the length with respect to $g$. Then my guess is as follows:

Question. Can we find a constant $c$ so that $$ \langle \alpha, [\gamma] \rangle \le c \|\alpha\|_g \cdot L_g(\gamma) $$ for any $\alpha$ and $[\gamma]$? If so, can we simply take $c=1$?