Surgery obstruction, signature (Ex)
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2) Prove that if then we can find a Lagrangian with respect to . (This is proven in [Milnor1961, Lemma 8 and 9 of Milnor1961]
You may assume the following lemma:
Lemma 0.1 [Milnor1961, Lemma 8]. A quadratic form with integer coefficients and determinant has a non-trivial zero if and only if it is indefinite.
References
- [Milnor1961] J. Milnor, A procedure for killing homotopy groups of differentiable manifolds, Proc. Sympos. Pure Math, Vol. III (1961), 39–55. MR0130696 (24 #A556) Zbl 0118.18601