Freedman Theorem

Two closed simply connected 4-manifolds are homeomorphic iff they have the same bilinear form  and the same Kirby-Siebenmann invariant . Any  can be realized by such a manifold. If  is odd for some , then either value of  can be realized also. However, if  is always even, then  is determined by , being congruent to 1/8 of the signature of . Here,  is a symmetric bilinear form with determinant  (Milnor).

In particular, if  is a homotopy sphere, then  and , so  is homeomorphic to .

REFERENCES:

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