Surgery obstruction, Arf-invariant (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")		Latest revision as of 21:18, 25 August 2013 (view source) Spiros Adams-Florou (Talk | contribs)
Line 1:		Line 1:
	<wikitex>;		<wikitex>;
−	...	+	Let $(K,\lambda,\mu)$ be a $(-1)$-symmetric unimodular quadratic form over $\Z$ so that $(K,\lambda)\cong H_{-1}(\Z^r)$ with canonical basis $\{e_1,\ldots, e_r,f_1,\ldots, f_r\}$. Recall that the quadratic refinement $\mu: K \to \Z_2$ is a function such that for all $x,y\in K$, $$\mu(x+y) = \mu(x)+\mu(y) + \lambda(x,y)\quad (\mathrm{mod}\;2).$$ Define the '''Arf invariant''' of $(K,\lambda,\mu)$ by $$A(K,\lambda,\mu) := \sum_{i=1}^r\mu(e_i)\mu(f_i) \in \Z_2.$$ Prove that the Arf invariant is well-defined and defines an isomorphism $$A:L_2(\Z)\cong \Z_2.$$
		+
		+	'''Hint:''' Start by classifying quadratic forms on $\Z^2$, use induction and also count the size of the sets $\mu^{-1}(0)$ and $\mu^{-1}(1)$.
	</wikitex>		</wikitex>
	== References ==		== References ==

---

Latest revision as of 21:18, 25 August 2013

Let $<wikitex>; Let $(K,\lambda,\mu)$ be a $(-1)$-symmetric unimodular quadratic form over $\Z$ so that $(K,\lambda)\cong H_{-1}(\Z^r)$ with canonical basis $\{e_1,\ldots, e_r,f_1,\ldots, f_r\}$. Recall that the quadratic refinement $\mu: K \to \Z_2$ is a function such that for all $x,y\in K$, $$\mu(x+y) = \mu(x)+\mu(y) + \lambda(x,y)\quad (\mathrm{mod}\;2).$$ Define the '''Arf invariant''' of $(K,\lambda,\mu)$ by $$A(K,\lambda,\mu) := \sum_{i=1}^r\mu(e_i)\mu(f_i) \in \Z_2.$$ Prove that the Arf invariant is well-defined and defines an isomorphism $$A:L_2(\Z)\cong \Z_2.$$ '''Hint:''' Start by classifying quadratic forms on $\Z^2$, use induction and also count the size of the sets $\mu^{-1}(0)$ and $\mu^{-1}(1)$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]](K,\lambda,\mu)$ be a $(-1)$-symmetric unimodular quadratic form over $\Z$ so that $(K,\lambda)\cong H_{-1}(\Z^r)$ with canonical basis $\{e_1,\ldots, e_r,f_1,\ldots, f_r\}$. Recall that the quadratic refinement $\mu: K \to \Z_2$ is a function such that for all $x,y\in K$, Define the Arf invariant of $(K,\lambda,\mu)$ by Prove that the Arf invariant is well-defined and defines an isomorphism

Hint: Start by classifying quadratic forms on $\Z^2$, use induction and also count the size of the sets $\mu^{-1}(0)$ and $\mu^{-1}(1)$.

[edit] References