Surgery obstruction, signature (Ex)

Revision as of 20:53, 25 August 2013 (view source) Diarmuid Crowley (Talk | contribs) (Created page with "<wikitex>; ... </wikitex> == References == {{#RefList:}} Category:Exercises Category:Exercises without solution")		Revision as of 21:32, 25 August 2013 (view source) Spiros Adams-Florou (Talk | contribs) Newer edit →
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−	...	+	'''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 $$\mathrm{sign}(K_n(M),\lambda) = \mathrm{sign}(H_n(M),\lambda_M) - \mathrm{sign}(H_n(X),\lambda_X).$$
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		+	'''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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		+	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}}
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	</wikitex>		</wikitex>
	== References ==		== References ==

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Revision as of 21:32, 25 August 2013

1) Let $<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 $n$-connected degree one normal map. Let $(K_n(M),\lambda, \mu)$ be the associated kernel form. Prove that signature satisfies the relation

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 [Milnor1961, Lemma 8 and 9 of Milnor1961]

You may assume the following lemma: {{beginthm|Lemma|[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. </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