Algebraic Topology II - 2019

Joel, 26.08.2019 9:30-10:00

Half an hour before the time on mystudies, you go into the preperation room where you can work on the problems, and also ask questions if necessary. After half an hour you go into the main exam, where you can write your prepared notes on the blackboard. It is a very relaxed atmosphere, and Biran also gives helpful comments when you're stuck.

My questions where:

1. Define Cohomology with compact supports.

2. Calculate \( H_c^*(\mathbb{R}^n;G) \).

3. Does a map \(f:X \rightarrow Y\) induce a map \(f^* : H_c^*(Y;G) \rightarrow H_c^*(Y;G)\) ?

4. Do homotopic spaces have the same cohomologies with cpt support?

5. Is the cup product well defined on cohomology with cpt support?

6. State the Poincaré duality theorem for (not neccesarrily cpt) manifolds.

7. Show that \(H_c^i(X \times \mathbb{R};G) = H_c^{i-1}(X;G) \).

Good Luck!

Daniel, 30.08.2019 11:00-12:00

1. Let \(f: \mathbb{R}P^2\rightarrow \mathbb{R}P^2/\mathbb{R}P^1 = S^2\) be the map that collapses \(\mathbb{R}P^1\subset \mathbb{R}P^2\) to a point. Calculate \(f^{\star}: H^2(S^2; \mathbb{Z}_2)\rightarrow H^2(\mathbb{R}P^2; \mathbb{Z}_2)\). Here \( \mathbb{R}P^2 = B^2/x\sim -x \ \forall x\in \partial B^2\), \(\mathbb{R}P^1 = \partial B^2/x\sim -x\ \forall x\in \partial B^2\).

2. Suppose we have two spaces X and Y, \(k\geq 0\) and an abelian group G such that \(\text{Ext}(H_{k-1}(X), G) = \text{hom}(H_k(Y), G) = 0.\) Is it true that \(f^{\star}: H^{k}(Y, G)\rightarrow H^{k}(X, G)\) is 0?

3. Is it true that the splitting in the universal coefficient theorem for cohomology is always natural?

4. Suppose M is a non-orientable, closed and connected manifold. Let \(\pi: M'\rightarrow M\) be the 2:1 orientation covering (so that M' is orientable). Let \(\theta: M'\rightarrow M'\) be the unique non trivial deck transformation. Show that \(\text{deg}(\theta) = -1\).