Can we say that the moduli of differentiable structures of a topological manifold is discrete?

Question

Differentiabl structure is defined by tangent bundle and any bundle determines a homotopy class of a map from the manifold to the Grassmanian manifold of the planes whose dimension is the rank of the tangent bundle. Because, I am not sure but..., I guess the homotoply class of such map is descrete group,... so I geuss that the differentiable structure is also discrete.

Answer 1

This paper constructs a certain moduli space of smooth structures on $\mathbb{R}^4$s which does not have the discrete topology, although note that it is a quotient of the "true moduli space" because some exotic non-diffeomorphic $\mathbb{R}^4$s are identified.

https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/moduli-space-of-exotic-4s/2921256CAE206372C7998F38D20DB082