# Fibre homotopy trivial bundles (Ex)

From Manifold Atlas

**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.