Fibre homotopy trivial bundles (Ex)

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Exercise 0.1.

  1. 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 \Zz/24, \pi_3(O_5) \cong \Zz and the fact that the J-homomophism in dimension 3 is isomorphic to the surjective homomorphism \Zz \to \Zz/24, find a homotopy equivalence of manifolds f \colon M_0 \simeq M_1 such that f^*p(M_0) \neq p(M_1). Here p(M_i) \in H^{4*}(M_i; \Zz) denotes the total Pontrjagin class.
  2. The above exercise showed that the first Pontrjain class, p_1, is not a homotopy invariant. Apply the same idea to show that p_k is not a homotopy invariant for any k \geq 1.

Remark 0.2. A reference to Novikov is needed here.

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