Here I write down the definitions and theorems I want to remember for the course Algebraic Topology:

Definition(Homotopy extension property) A subspace \(A\subseteq X\) of a topological \(X\) has the homotopy extension property if for all maps \(f,g:X\to Z\) with homotopy \(H:f|_{A}\implies g\), there exists an extension of \(H\), \(H^*: f\implies g^*\).

Theorem 2.16 If \(X\) is a non-empty top then \(H_0(X;A)\cong A[\pi_0(X)]\).

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.

2. 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.
3. 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 \(H\)*
4. 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

Definition An \(n\)-ary boolean function is a function \(f:\{0,1\}^n\to \{0,1\}\).

Proposition For every boolean function \(f\) there is a propositional formula \(\varphi\) such that the function defined by \(\varphi\) is equal to \(f\).

Definition The size of a circuit is defined as the amount of and-, not-, and or-gates.

Definition Suppose that \(\{C_n\}_{n\in \mathbb{N}}\) is a family of circuits such that for every \(n\), \(C_n\) computes a boolean function of \(n\) inputs. A family computes a language \(L\) if for all \(x\in \{0,1\}^n\) we have