Math and stuff

Proposition Let $0 \rightarrow M’ \rightarrow M \rightarrow M’’ \rightarrow 0$ be an exact sequence of $A$-modules. If $M’$ and $M’’$ are finitely generated, then so is $M$. Solution Let $i$ denote the map from $M’$ into $M$ and $p$ denote the map from $M$ to $M’’$. Let $\{ x_1,...

Proposition Let $\gamma$ be the unit circle. Find a Mobius transformation that transforms $\gamma$ onto $\gamma$ and transforms $0$ to $1/2$. Solution We will use the formula given here. $f(z) = (z - 1/2) / (1 - z/2)$ maps $\gamma$ onto $\gamma$ and sends 1/2 to 0. It suffices to...

Proposition Suppose \[\begin{align*} A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{align*}\] is a $2 \times 2$ matrix of complex numbers whose determinant $ad - bc$ is nonzero. Then we can define a corresponding Mobius transformation on $\hat{\mathbb{C}}$ by $T_A(z) = \frac{az + b}{cz + d}$. Show...

Proposition Find Mobius transformations satisfying each of the following. Write your answers in standard form, as $\frac{az + b}{cz + d}$. $1 \mapsto 0, 2 \mapsto 1, 3 \mapsto \infty$. $1 \mapsto 0, 1 + i \mapsto 1, 2 \mapsto \infty$. $0 \mapsto i, 1 \mapsto 1, \infty \mapsto -i$....

Proposition Find a counterexample to Theorem 5-2 if condition (3) is omitted. [Spivak] Solution Consider the following manifold which can be obtained from wrapping $(0, 1)$. For any $x \in M$, let $U = \mathbb{R}^2$ and $W = (0, 1) \subset \mathbb{R}^1$. Then the function $f(t) = (f_1(t), f_2(t))$ which...