Let M M be a connected sum of r r closed aspherical manifolds of dimension n ⩾ 3 n \geqslant 3 , and let E M EM denote the space of self-homotopy-equivalences of M M , with basepoint the identity map of M M . Using obstruction theory, we calculate π q ( E M ) {\pi _q}(EM) for 1 ⩽ q ⩽ n − 3 1 \leqslant q \leqslant n - 3 and show that π n − 1 ( E M ) {\pi _{n - 1}}(EM) is not finitely-generated. As an application, for the case n = 3 n = 3 and r ⩾ 3 r \geqslant 3 we show that infinitely many generators of π 1 ( E M 3 , i d M ) {\pi _1}(E{M^3},{\text {i}}{{\text {d}}_M}) can be realized by isotopies, to conclude that π 1 ( Homeo ( M 3 ) , i d M ) {\pi _1}({\text {Homeo}}({M^3}),{\text {i}}{{\text {d}}_M}) is not finitely-generated.