Splitting invariants (Ex)

Revision as of 14:22, 23 March 2012 (view source) Marek Kaluba (Talk | contribs) (Created page with "<wikitex>; Prove that two maps $f_1,f_2 \colon \mathbb{C}P^n \to G/PL$ are homotopic iff their splitting invariants agree for $2 \leq i \leq n$. Use the exact sequence $$L_{2...")		Latest revision as of 15:00, 1 April 2012 (view source) Diarmuid Crowley (Talk | contribs) m
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	[\mathbb{C}P^{k-1},G/PL] \to 0$$		[\mathbb{C}P^{k-1},G/PL] \to 0$$
	and the fact that the surgery obstruction map $$\theta \colon		and the fact that the surgery obstruction map $$\theta \colon
−	[\mathbb{C}P^k, G/PL] \to L_{2k}(\mathbb{Z})$$ splits the above sequence for $k>2$. Additionally $[\mathbb{C}P^2,G/PL]=\mathbb{Z}$ and the isomorphism is given by the surgery obstruction map.	+	[\mathbb{C}P^k, G/PL] \to L_{2k}(\mathbb{Z})$$ splits the above sequence for $k>2$. Additionally $[\mathbb{C}P^2,G/PL] \equiv \mathbb{Z}$ and the isomorphism is given by the surgery obstruction map.
	</wikitex>		</wikitex>
−	== References ==	+	<!-- == References ==
−	{{#RefList:}}	+	{{#RefList:}} -->
	[[Category:Exercises]]		[[Category:Exercises]]
		+	[[Category:Exercises without solution]]

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Latest revision as of 15:00, 1 April 2012

Prove that two maps $<wikitex>; Prove that two maps $f_1,f_2 \colon \mathbb{C}P^n \to G/PL$ are homotopic iff their splitting invariants agree for \leq i \leq n$. Use the exact sequence $$L_{2k}(\mathbb{Z}) \to [\mathbb{C}P^k,G/PL] \to [\mathbb{C}P^{k-1},G/PL] \to 0$$ and the fact that the surgery obstruction map $$\theta \colon [\mathbb{C}P^k, G/PL] \to L_{2k}(\mathbb{Z})$$ splits the above sequence for $k>2$. Additionally $[\mathbb{C}P^2,G/PL]=\mathbb{Z}$ and the isomorphism is given by the surgery obstruction map. </wikitex> == References == {{#RefList:}} [[Category:Exercises]]f_1,f_2 \colon \mathbb{C}P^n \to G/PL$ are homotopic iff their splitting invariants agree for $2 \leq i \leq n$.

Use the exact sequence

and the fact that the surgery obstruction map splits the above sequence for $k>2$. Additionally $[\mathbb{C}P^2,G/PL] \equiv \mathbb{Z}$ and the isomorphism is given by the surgery obstruction map.