Mountain pass energies between homotopy classes of maps

For non-homotopic maps $u,v\in C^{\infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $\gamma_p(u,v)$ for which there exist paths $u_t\in W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $\|du_t\|_{L^p}^p\leq \gamma_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows that $\gamma_p(u,v)<\infty$ for $p\in [1,k)$, and using their construction, one can obtain an estimate of the form $\gamma_p(u,v)\leq \frac{C(u,v)}{k-p}$. When $M$ and $N$ are oriented, and $u$ and $v$ induce different maps on real cohomology in degree $k-1$, we show that the growth $\gamma_p(u,v)\sim \frac{1}{k-p}$ as $p\to k$ is sharp, and obtain a lower bound for the coefficient $\liminf_{p\to k}(k-p)\gamma_p(u,v)$ in terms of the min-max masses of certain non-contractible loops in the space of codimension-$k$ integral cycles in $M$. In the process, we establish lower bounds for a related smaller quantity $\gamma_p^*(u,v)\leq\gamma_p(u,v)$, for which there exist critical points $u_p\in W^{1,p}(M,N)$ of the $p$-energy functional satisfying $\gamma_p^*(u,v)\leq \|du_p\|_{L^p}^p\leq \gamma_p(u,v).$