Algebraic Topology II - 2018
Please sign with your name and the date on which you had your exam. If you use this wiki, contribute to it as well or terrible things will happen to you: like me kicking you with my fists.
Moji Hosseinzadeh, 22.08.2018, 11:00-11:30
Prof. Merry asked be whether I want to write on paper or on the blackboard and for convenience I chose paper. Then I could draw my exam sheet out of his hand and had the following
Proof Question 1
- Show that the tensor product satisfies the UP (lemma 24.4)
- Give the definition of a right-exact additive functor and show that tensoring with A is a right-exact functor (Prop. 24.10)
Proof Question 2
- Definition of a weak fibration
- Prove Serre's theorem (Thm 45.16)
Problem Question
- This was about generalising the degree to manifolds - explicitly showing that a map f from your manifold M to B^n/S^{n-1} which is isomorphic to S^n has degree ±1 and then as a second part show, that one can construct maps of degree k for k in Z by composing it with a map g from S^n to S^n, which has degree k.
In general Prof. Merry and Berit are very kind and help a lot when you get stuck. For the Problem Question hand-wavy arguments were fine.
Lukas, 23.08.2018, 13:00-13:30
Prof. Merry asked be whether I want to write on paper or on the blackboard and for convenience I chose paper. Then I could draw my exam sheet out of his hand and had the following
Proof Question 1
- Define \( \operatorname{Tor}(A,B)\), show that it is well-defined.
- State the UCT, use it to show \( \operatorname{H}_n(X;\mathbb{Q}) = \operatorname{H}_n(X)\otimes \mathbb{Q} \)
Proof Question 2
- forgot (sorry!)
Problem Question
- This was about the Euler characteristic \( \chi(M) := \sum_{i=0}^n (-1)^i \operatorname{rank}H_i(M)\). First, show that \(\chi(M) = \sum_{i=0}^n (-1)^i \operatorname{dim} H_i(M;\mathbb{Z}_2)\). Then, show that for a closed oriented topological manifold of dimension \(n = 2m+1\) we have that \(\chi (M) = 0\).
In general Prof. Merry did not say very much, he only gave short hints on what I should look at next.
