The exact statement of the conjecture is as follows: a closed 4-dimensional smooth manifold which is homotopy equivalent to the 4-sphere is diffeomorphic to the standard 4-sphere.

There is an equivalent statement in the category of piecewise linear manifolds (i.e. if we change "smooth" to "piecewise linear" and change "diffeomorphic" to "PL-isomorphic").

An analogous statement in the category of topological manifolds (i.e. remove the "smooth" assumption and change "diffeomorphic" to "homeomorphic") is known to be true.

For more details on the conjecture, see: https://en.wikipedia.org/wiki/Generalized_Poincar%C3%A9_conjecture