Splitting invariants (Ex)
From Manifold Atlas
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[\mathbb{C}P^k, G/PL] \to L_{2k}(\mathbb{Z})$$ splits the above sequence for $k>2$. Additionally $[\mathbb{C}P^2,G/PL] \equiv \mathbb{Z}$ and the isomorphism is given by the surgery obstruction map. | [\mathbb{C}P^k, G/PL] \to L_{2k}(\mathbb{Z})$$ 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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[[Category:Exercises]] | [[Category:Exercises]] | ||
+ | [[Category:Exercises without solution]] |
Latest revision as of 15:00, 1 April 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.