Surgery obstruction, signature (Ex)
From Manifold Atlas
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| − | '''1)''' Let $(\overline{f} | + | '''1)''' Let $(f, \overline{f}):M^{2n} \to X$ be an $n$-connected degree one normal map. Let $(K_n(M),\lambda, \mu)$ be the associated kernel form. Prove that signature satisfies the relation $$\mathrm{sign}(K_n(M),\lambda) = \mathrm{sign}(H_n(M),\lambda_M) - \mathrm{sign}(H_n(X),\lambda_X).$$ |
'''2)''' Prove that if $\mathrm{sign}(K_n(M),\lambda)=0 $ then we can find a Lagrangian with respect to $\lambda$. (This is proven in \cite{Milnor1961|Lemma 8 and 9 of Milnor1961} | '''2)''' Prove that if $\mathrm{sign}(K_n(M),\lambda)=0 $ then we can find a Lagrangian with respect to $\lambda$. (This is proven in \cite{Milnor1961|Lemma 8 and 9 of Milnor1961} | ||
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A quadratic form $\varphi$ with integer coefficients and determinant $\pm 1$ has a non-trivial zero if and only if it is indefinite. | A quadratic form $\varphi$ with integer coefficients and determinant $\pm 1$ has a non-trivial zero if and only if it is indefinite. | ||
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== References == | == References == | ||
Latest revision as of 01:38, 6 January 2019
be an 
-connected degree one normal map. Let 
be the associated kernel form. Prove that signature satisfies the relation 
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
