Theorem 32.1:

Let $X$ be a path connected topological space. There is a surjective group homomorphism

$$\Pi_X:\pi_1(X, x_0)\longrightarrow H_1(X) \eqno(32.2)$$

whose kernel is the commutator subgroup $[\pi_1(X, x_0), \pi_1(X, x_0)]$. Thus

$$H_1(X) = \pi_1(X, x_0)/[\pi_1(X, x_0), \pi_1(X, x_0)] \eqno(32.3)$$

Before taking up the proof which will be completed in several steps, we set up the map $\Pi_X$. Note that if $\gamma$ is a loop in $X$ based at $x_0$ then $\gamma$ is a one cycle, that is to say $\gamma\in Z_1(X)$ and we denote its homology class in the quotient $H_1(X)$ by $\overline{\gamma}$. This suggests that we define $\Pi_X:\pi_1(X, x_0)\longrightarrow H_1(X)$ as

$$\Pi_{X} : [\gamma] \mapsto \overline\gamma \eqno(32.4)$$

We shall show that the map is a well-defined surjective group homomorphism and determine its kernel. We do each of these as a separate lemma. Since homotopy of loops is a map from the square $[0, 1]\times [0, 1]$ whereas a singular two simplex is a map from $\Delta_2$ to $X$ we must first set up some standard maps from $\Delta_2$ to the square with specific properties. The usual proofs seem slightly tricky and we shall try an approach that would be useful in the next lecture.

Divide the square $[0, 1]\times [0, 1]$ into two triangles by drawing a diagonal from $(0,0)$ to $(1, 1)$. Let $T_i$ ($i = 1, 2$) be two affine homeomorphisms mapping $\Delta_2$ onto the these two triangles given by

$$T_1({\widehat{\bf e}}_1) = (0, 0),\quad T_2({\widehat{\bf e}}_1) = (0, 0),$$

$$T_1({\widehat{\bf e}}_2) = (1, 0),\quad T_2({\widehat{\bf e}}_2) = (0, 1),$$

$$T_1({\widehat{\bf e}}_3) = (1, 1),\quad T_2({\widehat{\bf e}}_3) = (1, 1).$$

in We shall regard the maps $T_i$ ($i = 1, 2$) as maps from $\Delta_2$ into $I^2$ and use them to prove the following: