Fibre homotopy trivial bundles (Ex)

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Exercise 0.1.

1. Observe that $\pi_i(G_{k+1})$$; {{beginthm|Exercise}} # Observe that \pi_i(G_{k+1}) classifies k-spherical fibrations over S^{i+1}. Using the isomorphisms \pi_3(G_5) \cong \pi_3^S \cong \Zz/24, \pi_3(O_5) \cong \Zz and the fact that the J-homomophism in dimension is isomorphic to the surjective homomorphism \Zz \to \Zz/24, find a homotopy equivalence of manifolds f \colon M_0 \simeq M_1 such that f^*p(M_0) \neq p(M_1). Here p(M_i) \in H^{4*}(M_i; \Zz) denotes the total Pontrjagin class. # The above exercise showed that the first Pontrjain class, p_1, is not a homotopy invariant. Apply the same idea to show that p_k is not a homotopy invariant for any k \geq 1. {{endthm}} {{beginrem|Remark}} A reference to Novikov is needed here. {{endrem}} == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises with solution]]\pi_i(G_{k+1})$ classifies $k$$k$-spherical fibrations over $S^{i+1}$$S^{i+1}$. Using the isomorphisms $\pi_3(G_5) \cong \pi_3^S \cong \Zz/24$$\pi_3(G_5) \cong \pi_3^S \cong \Zz/24$, $\pi_3(O_5) \cong \Zz$$\pi_3(O_5) \cong \Zz$ and the fact that the J-homomophism in dimension $3$$3$ is isomorphic to the surjective homomorphism $\Zz \to \Zz/24$$\Zz \to \Zz/24$, find a homotopy equivalence of manifolds $f \colon M_0 \simeq M_1$$f \colon M_0 \simeq M_1$ such that $f^*p(M_0) \neq p(M_1)$$f^*p(M_0) \neq p(M_1)$. Here $p(M_i) \in H^{4*}(M_i; \Zz)$$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$$p_1$, is not a homotopy invariant. Apply the same idea to show that $p_k$$p_k$ is not a homotopy invariant for any $k \geq 1$$k \geq 1$.

Remark 0.2. A reference to Novikov is needed here.