Can an isomorphism induce a differentiable manifold structure

Question

I was thinking If defining a map from one topological space X to a set Y can induce topology on the set Y and make it a topological space then can the same be said for manifolds ? especially if i have an isomorphism between a space that has a manifold structure and a space that's unknown, can i say it will have a manifold structure? and will be of the same class, for example if X is isomorphic to Y and X has a differentiable manifold then will Y have a differentiable manifold structure as well?