Math 442 Final Exam Let Ξ = {P : E(Ξ) → B} Be a Euclidean R N Bundle

Math 442 Final Exam

Let ξ = {p : E(ξ) → B} be a Euclidean Rn bundle. Define the disk bundle p : D(ξ) → B as the vectors of length less than or equal to one in E(ξ). Then for all b ∈ B, the fiber p−1(b) is homeomorphic to the unit disk in Rn. Similarly, define the sphere bundle p : S(ξ) → B to be the vectors of length one in E(ξ); now the n−1 fiber is homeomorphic to S . The inclusions D(ξ) ⊆ E(ξ) and S(ξ) ⊆ E(ξ)0 are homotopy equivalences. Define the Thom space by T (ξ) = D(ξ)/S(ξ).

1. Suppose that ξ is R-oriented. Use the Thom isomorphism theorem to construct a Thom class U ∈ Hn(T (ξ),R) and an isomorphism Φ : Hk(B; R) → H˜ n+k(T (ξ); R).

2. Now suppose that ξ is Z-oriented and e(ξ) is the Euler class. Show that U ^ U = Φ(e(ξ)).

In particular, show that if n is odd then the Euler class has order 2.

3. Let τ be the tangent bundle of Sn. Show that T (τ) is a CW complex with a 0-cell, an n-cell, and a 2n-cell. Prove that if n is even then the attaching map S2n−1 → Sn of the 2n-cell has Hopf invariant 2.

4. Let γ be the canonical line bundle over CP n. Show that the Thom space T (γ) has the same integral cohomology rings as CP n+1. (It’s actually homeomorphic to CP n+1, but that’s not what I’m asking.) Thus

∗ n+2 H (T (γ), Z) = Z[U]/(U ). 5. Show that Alexander Duality and the Thom isomorphism theorem imply Poincaréduality for differentiable manifolds. (This is an observation of Atiyah.) Recall that Alexander duality says that if K ⊆ SN then (under appropriate hypotheses) k N ∼ H (S − K; R) = HN−k−1(K; R). Now let M ⊆ RN be a differentiable embedding. Now use the tubular neigh- borhood theorem. Our proof of Alexander duality used Poincaréduality; however, it is possible to give an independent (and easier) proof.

Over

1 In the remaining problems we will use Z/2Z = F2 coeﬃcients.

n n If ξ = {p : E → B} is an R bundle and U ∈ H (E,E0) is the unique n n F2 orientation, let wn(ξ) ∈ H (B) be the image of U under the H (E,E0) → HnE =∼ HnB. As the Euler class of a complex bundle is the top Chern class, this is the top Stiefel-Whitney class.

6. Let γ be the M¨obius (or canonical line) bundle over S1. Show directly 1 1 that w1(γ) is the generator of H (S ). On the other hand, show that wn(ξ) = 0 if ξ is the trivial bundle.

7. State and prove the existence of a mod 2 Gysin sequence, valid for an Rn-bundle.

8. Let γ be the canonical line bundle over RP n. Prove S(γ) = {Sn → RP n} is the standard quotient map. Use this and the Gysin sequence to give an new proof that

∗ n n+1 H RP = F2[w1(γ)]/(w1(γ) ], 1 ≤ n ≤ ∞.