Smale Theorem

If  is a differentiable homotopy sphere of dimension , then  is homeomorphic to . In fact,  is diffeomorphic to a manifold obtained by gluing together the boundaries of two closed -balls under a suitable diffeomorphism (Milnor).

REFERENCES:

Milnor, J. "The Poincaré Conjecture." https://www.claymath.org/millennium/Poincare_Conjecture/Official_Problem_Description.pdf.

Smale, S. "Generalized Poincaré's Conjecture in Dimensions Greater than Four." Ann. Math. 74, 391-406, 1961.