Splitting invariants (Ex)
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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 $$L_{2...") |
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[\mathbb{C}P^{k-1},G/PL] \to 0$$ | [\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{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]] |
Revision as of 14:46, 23 March 2012
Prove that two maps are homotopic iff their splitting invariants agree for .
Use the exact sequence
and the fact that the surgery obstruction map
splits the above sequence for . Additionally and the isomorphism is given by the surgery obstruction map.