Fibre homotopy trivial bundles (Ex)
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(Created page with "<wikitex>; {{beginthm|Exercise}} # Observe that $\pi_i(G_{k+1})$ classifies $k$-spherical fibrations over $S^{i+1}$. Using the isomorphisms $\pi_3(G_5) \cong \pi_3^S \cong \Z...")
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(Created page with "<wikitex>; {{beginthm|Exercise}} # Observe that $\pi_i(G_{k+1})$ classifies $k$-spherical fibrations over $S^{i+1}$. Using the isomorphisms $\pi_3(G_5) \cong \pi_3^S \cong \Z...")
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Revision as of 02:21, 28 May 2012
Exercise 0.1.
- Observe that classifies -spherical fibrations over . Using the isomorphisms , and the fact that the J-homomophism in dimension is isomorphic to the surjective homomorphism , find a homotopy equivalence of manifolds such that . Here denotes the total Pontrjagin class.
- The above exercise showed that the first Pontrjain class, , is not a homotopy invariant. Apply the same idea to show that is not a homotopy invariant for any .
Remark 0.2. A reference to Novikov is needed here.