Two closed simply connected 4-manifolds are homeomorphic iff they have the same bilinear form β and the same Kirby-Siebenmann invariant κ. Any β can be realized by such a manifold. If β(x⊗x) is odd for some x element H^2, then either value of κ can be realized also. However, if β(x⊗x) is always even, then κ is determined by β, being congruent to 1/8 of the signature of β. Here, β:H^2 ⊗H^2->H^4 ≅Z is a symmetric bilinear form with determinant ± 1 (Milnor). In particular, if M^4 is a homotopy sphere, then H^2 = 0 and κ = 0, so M^4 is homeomorphic to S^4.

Kirby-Siebenmann invariant | Poincaré conjecture | Smale theorem