Exam_2024 2025

The exercise for this exam that I should look at are: 1.2, 1.3, 1.4, 3.1, 3.2, 4, 5.2, 5.3. My answers are in italics.

Problem 1

For each of the following, decide whether they are true or false. Give a short argument or example justifying your answer.

  1. No CW-complex with infinitely many cells is contractible. The statement is true. As an example of this I would give \(S^\infty\) which is not contractible because of an exercise.
  2. The inclusion \(\{0\}\subseteq [0,1]\) has the homotopy extension property. This is true. Given a map \(f:[0,1]\to Z\) a homotopy \(H\) starting from \(f|_{\{0\}}\) is simply a path starting in \(f(0)\) in \(Z\). Then we can simply define \(g:[0,1]\to Z\) as \(g(t)=H(1)\). We define simply define |(H^:f\implies g) for \(x\in [0,1]\) as the path going from \(f(x)\) to \(f(0)\) concatenated with the path given by \(H\)*
  3. Let \(X\to Y\) be an injective map. Then \(H_k(X; Z) \to H_k(Y; Z) is also injective for all \(k \geq 0\). *This is definitely not the case. If we choose \(X\) such that it has two path connected components an \(Y\) to be a superspace with only one path component, then \(H_k(X;\mathbb{Z})\cong \mathbb{Z}\otimes \mathbb{Z}\) and (H_k(Y;\mathbb{Z})\cong \mathbb{Z}) which does not allow an injection.

Problem 3

Let \(\mathbb{C}P^n\) denote complex projective space; recall that this can also be defined as the quotient of \(S^{2n+1}\) by the equivalence relation \(w \sim \lambda w\) for \(w \in S^{2n+1}\subseteq \mathbb{C}^{n+1}\) and \(\lambda \in S^1\subseteq \mathbb{C}\).

  1. Show that \(\mathbb{C}P^n\) is homeomorphic to the cell attachment \(\mathbb{C}P^{n-1}\sqcup_p D^{2n}\), where \(p:S^{2n-1}\to \mathbb{C}P^{n-1}\) is the quotient map. You may also use without proof that
\[\mathbb{C}P^n\cong D^{2n}/\sim\]

where \(sim\) identifies \(w\) with \(\lambda w\) for each \(w\in S^{2n-1}\subseteq \mathbb{C}^n\) and \(\lambda \in S^1 \subseteq \mathbb{C}\).

*In the case \(n=0\) we have \(\mathbb{C}P^n\) is the one point space, \(D^{2n\}\) is the one point sapce and \(\mathbb{C}P^{n-1}\) is the zero point space, hence \(\mathbb{C}P^{n-1}\sqcup_p \mathbb{C}P^n\) is also the one point space, thus they are clearly homeomorphic.

Now for \(n>0\) we get that

\[\(\mathbb{C}P^n\) \cong D^{2n}/\sim \cong\]

*

I have decided now that doing everything on the computer takes to much time. I will do the rest of the exam on paper.