Poincare Duality

Let \(X\) be a connected compact orientable manifold of dimension \(n\). Explain carefully how Poincaré duality implies the existence of a bilinear pairing \[ H^k(X;\mathbb {Z})\times H^{n-k}(X;\mathbb {Z})\to \mathbb {Z} \] that modulo torsion becomes a perfect (i.e., non-singular) pairing. Deduce from this the structure of the cohomology ring \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\).

1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
2. What is \(H^k(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give a proof by induction on \(n\) using (a). State any theorems you are using.
3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n;\mathbb {Z})\)? Give an inductive proof. You may use Poincaré duality.

1. The complex projective space \(\mathbb {C}\mathrm {P}^n\) can be obtained from \(\mathbb {C}\mathrm {P}^{n-1}\) by attaching a single \(2n\)-cell. Describe the attaching map.
2. What is \(H^k(\mathbb {C}\mathrm {P}^n)\) (integral coefficients)? Give a proof by induction on \(n\) using (a).
3. What is the ring structure on \(H^*(\mathbb {C}\mathrm {P}^n)\)? Give an inductive proof. You may use Poincaré duality.

Let \(M\) be a connected topological \(n\)-manifold. Define what it means for \(M\) to be orientable. Then state the Poincaré duality theorem (including a careful definition of the map that is claimed to be an isomorphism).