Lower bounds are found for linear and Alexandrov's cowidths of Sobolev's classes on compact homogeneous Riemannian manifolds \(M^d\). Using these results, the authors give an explicit solution for the problem of optimal reconstruction of functions from Sobolev's classes \(W_p^r(M^d)\) in \(L_q(M^d)\), \(1 \leq q \leq p\leq \infty\). In particular, for the Gelfand widths they get \[ d^n (W_p^r(M^d), L_q(M^d)) \gg \begin{cases} n^{-r/d},& 1