Splitting invariants (Ex)

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

Use the exact sequence

$\displaystyle L_{2k}(\mathbb{Z}) \to [\mathbb{C}P^k,G/PL] \to [\mathbb{C}P^{k-1},G/PL] \to 0$
and the fact that the surgery obstruction map
$\displaystyle \theta \colon [\mathbb{C}P^k, G/PL] \to L_{2k}(\mathbb{Z})$
splits the above sequence for $k>2$$k>2$. Additionally $[\mathbb{C}P^2,G/PL]=\mathbb{Z}$$[\mathbb{C}P^2,G/PL]=\mathbb{Z}$ and the isomorphism is given by the surgery obstruction map.