Splitting invariants (Ex)

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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

\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. Additionally [\mathbb{C}P^2,G/PL] \equiv \mathbb{Z} and the isomorphism is given by the surgery obstruction map.
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