Exercises

1. Show that the sphere $S^3$ is isomorphic (as a topological group) to $SU(2,\mathbb{C})$.
2. Show that the center of the group of non-zero quaternions is the set of non-zero real numbers. In the light of this explain why ker$D\psi(1)$ in lemma (22.6) is non-trivial.
3. Explain why the map $\phi$ defined in theorem (22.8) is bijective.
4. Verify the properties of the map $T_A$ in the proof of theorem (22.10). Also fill in the details concerning the properties of the map $\phi$ (except for the claims made concerning its derivative).
5. Use exercise 4 to find a generator of $\pi_1(SO(3, \mathbb{R}))$. Let $i:SO(2,\mathbb{R})\longrightarrow SO(3, \mathbb{R})$ be given by
   $$A \mapsto \begin{pmatrix} A & 0 \cr 0 & 1 \cr \end{pmatrix}, \quad A \in SO(2, \mathbb{R}).$$
   Show that $i_*:\pi_1(SO(2, \mathbb{R}))\longrightarrow \pi_1(SO(3, \mathbb{R}))$ is surjective.

in

Lectures - XXIII and XXIV Coproducts and Pushouts

in We now discuss further categorical constructions that are essential for the formulation of the Seifert Van Kampen theorem. We first discuss the notion of coproducts which is a prerequisite for a proof of the existence of push-outs. The coproduct is popularly known as the free product in the context of groups but we shall also use the term coproduct which seems more appropriate from a categorical point of view ([11], p. 71). The notion of coproducts has already been introduced in the exercises to lecture 7 for the categories Top and AbGr where it is popularly known as the disjoint union and the direct sum respectively. However the construction is more complicated in the category Gr. The coproduct is defined in terms of a universal property.