Freedman Theorem

Two closed simply connected 4-manifolds are homeomorphic iff they have the same bilinear form beta and the same Kirby-Siebenmann invariant kappa. Any beta can be realized by such a manifold. If beta(x tensor x) is odd for some x in H^2, then either value of kappa can be realized also. However, if beta(x tensor x) is always even, then kappa is determined by beta, being congruent to 1/8 of the signature of beta. Here, beta:H^2 tensor 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 kappa=0, so M^4 is homeomorphic to S^4.

See also

Kirby-Siebenmann Invariant, Poincaré Conjecture, Smale Theorem

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Milnor, J. "The Poincaré Conjecture."

Referenced on Wolfram|Alpha

Freedman Theorem

Cite this as:

Weisstein, Eric W. "Freedman Theorem." From MathWorld--A Wolfram Web Resource.

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