For non-homotopic maps $u,vin C^{infty}(M,N)$ between closed Riemannian manifolds, we consider the smallest energy level $gamma_p(u,v)$ for which there exist paths $u_tin W^{1,p}(M,N)$ connecting $u_0=u$ to $u_1=v$ with $|du_t|_{L^p}^pleq gamma_p(u,v)$. When $u$ and $v$ are $(k-2)$-homotopic, work of Hang and Lin shows that $gamma_p(u,v)<infty$ for $pin [1,k)$, 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 $pto k$ is sharp, and obtain a lower bound for the coefficient $liminf_{pto k}(k-p)gamma_p(u,v)$ 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 $gamma_p^*(u,v)leqgamma_p(u,v)$, for which there exist critical points $u_pin W^{1,p}(M,N)$ of the $p$-energy functional satisfying $gamma_p^*(u,v)leq |du_p|_{L^p}^pleq gamma_p(u,v).$
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