Math and stuff

Proposition Suppose $X$ is a topological space and $Y$ is an open subset of $X$. Show that the collection of all open subsets of $X$ that are contained in $Y$ is a topology on $Y$. Solution Let $\mathcal{T}$ be the topology on $X$, and $\mathcal{T}’ = \{ U \in \mathcal{T}...

Proposition Construct an explicit deformation retraction of the torus with one point deleted onto a graph consisting of two circles intersecting in a point, namely, longitude and meridian circles of the torus. Solution First, we will define a torus. Let $X = \{ (x, y) \in \mathbb{R}^2 \mid \abs{x} \leq...

Let $\{ G_{\alpha} \}$ be a collection of groups. We will define the free product \(*_{\alpha} G_{\alpha}\). Definition of a word Let $m$ be a non-negative integer. Let $G_{\alpha_1}, \cdots, G_{\alpha_m}$ be given. A function $f: \{ 1, 2, \cdots, m \} \rightarrow G_{\alpha_1} \times \cdots \times G_{\alpha_m}$ is called...

Proposition Show that every homomorphism $\pi_1(S^1) \rightarrow \pi_1(S^1)$ can be realized as the induced homomorphism \(\phi_*\) of a map $\phi: S^1 \rightarrow S^1$. Solution Instead of $\mathbb{R}^2$, we will regard the plane as $\mathbb{C}$. Then the loop $f(t) = e^{2\pi it}$ is a generator by Theorem 1.7 (Hatcher). Let a...

Proposition Construct an explicit deformation retraction of $\mathbb{R}^n \setminus \{ 0 \}$ onto $S^{n - 1}$. Solution Let $F: \mathbb{R}^n \setminus \{ 0 \} \times I \rightarrow S^{n - 1}$ be defined such that \[\begin{align*} F((x_1, \cdots, x_n), t) = \frac{(x_1, \cdots, x_n)}{(1 - t) + td} \end{align*}\] where $d...