Surgery obstruction, signature (Ex)

1) Let $(\overline{f},f):M^{2n} \to X$$<wikitex>; '''1)''' Let $(\overline{f},f):M^{2n} \to X$ be an $n$-connected degree one normal map. Let $(K_n(M),\lambda, \mu)$ be the associated kernel form. Prove that signature satisfies the relation $$\mathrm{sign}(K_n(M),\lambda) = \mathrm{sign}(H_n(M),\lambda_M) - \mathrm{sign}(H_n(X),\lambda_X).$$ '''2)''' Prove that if $\mathrm{sign}(K_n(M),\lambda)=0 $ then we can find a Lagrangian with respect to $\lambda$. (This is proven in \cite{Milnor1961|Lemma 8 and 9 of Milnor1961} You may assume the following lemma: {{beginthm|Lemma|\cite{Milnor1961|Lemma 8}} A quadratic form $\varphi$ with integer coefficients and determinant $\pm 1$ has a non-trivial zero if and only if it is indefinite. {{endthm}} </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]](\overline{f},f):M^{2n} \to X$ be an $n$$n$-connected degree one normal map. Let $(K_n(M),\lambda, \mu)$$(K_n(M),\lambda, \mu)$ be the associated kernel form. Prove that signature satisfies the relation

$\displaystyle \mathrm{sign}(K_n(M),\lambda) = \mathrm{sign}(H_n(M),\lambda_M) - \mathrm{sign}(H_n(X),\lambda_X).$ $$\displaystyle \mathrm{sign}(K_n(M),\lambda) = \mathrm{sign}(H_n(M),\lambda_M) - \mathrm{sign}(H_n(X),\lambda_X).$$

2) Prove that if $\mathrm{sign}(K_n(M),\lambda)=0$$\mathrm{sign}(K_n(M),\lambda)=0$ then we can find a Lagrangian with respect to $\lambda$$\lambda$. (This is proven in [Milnor1961, Lemma 8 and 9 of Milnor1961]

You may assume the following lemma: {{beginthm|Lemma|[Milnor1961, Lemma 8]} A quadratic form $\varphi$$\varphi$ with integer coefficients and determinant $\pm 1$$\pm 1$ has a non-trivial zero if and only if it is indefinite. </div>

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