Exercises

1. Prove theorem (31.3).
2. Show that for a path connected space $X$, every singleton $\{p\}$ with $p \in X$ is a basis for $H_0(X)$.
3. Complete the proof of theorem (31.5).
4. Show that the set ${\bf v}_1, {\bf v}_2, \dots, {\bf v}_k \in \mathbb{R}^n$ is affinely independent if the vectors
   $${\bf v}_1 - {\bf v}_{j},\dots,{\bf v}_{j-1} - {\bf v}_j, {\bf v}_{j+1} - {\bf v}_j, \dots, {\bf v}_{k-1} - {\bf v}_k$$
   are linearly independent for any $j$ ( $1\leq j\leq k$).
5. Prove theorem (31.6). Show that the barycentric coordinates are continuous functions of ${\bf x}$. All but the $j$-th barycentric coordinates of ${\bf v}_j$ vanish. The set of points in (31.2) obtained by setting $t_j = 0$ and varying the other coefficients is called the $j-$th face of the simplex spanned by the given points.
6. Check the continuity of the map $T\sigma$ in theorem (31.7).

in

Lecture - XXXII The abelianization of the fundamental group

in In this lecture we shall establish a basic result relating the fundamental group $\pi_1(X, x_0)$ and the first homology group $H_1(X)$. The result is elegant and states that $H_1(X)$ is the abelianization of $\pi_1(X, x_0)$ when $X$ is a path connected space. Further, the abelianization map is natural in the following sense. Suppose that $f: (X, x_0) \longrightarrow (Y, y_0)$ is a continuous map we have the following commutative diagram:

$\begin{CD} \pi_1(X, x_0) @> {f}_{*} >> \pi_1(Y, y_0) \\ @V{\Pi_X}VV @VV{\Pi_Y} V\\ H_1(X) @> H_1(f) >> H_1(Y) \\ \end{CD}$(32.1)

where $\Pi_X:\pi_1(X, x_0)\longrightarrow H_1(X)$ and $\Pi_Y:\pi_1(Y, y_0)\longrightarrow H_1(Y)$ are the quotient maps onto the respective abelianizations. We shall prove the main theorem (32.1) through a series of lemmas.