Example 1.1.0.2 (0006): The Fundamental Group

Example 1.1.0.2 (The Fundamental Group). Let $X$ be a topological space equipped with a base point $x \in X \simeq \operatorname{Sing}_0(X)$. Then continuous paths $p: [0,1] \rightarrow X$ satisfying $p(0) = x = p(1)$ can be identified with elements of the set $\{ \sigma \in \operatorname{Sing}_1(X): d^{1}_0(\sigma ) = x = d^{1}_1(\sigma ) \} $. The fundamental group $\pi _1(X,x)$ can then be described as the quotient

\[ \{ \sigma \in \operatorname{Sing}_1(X): d^{1}_0(\sigma ) = x = d^{1}_1(\sigma ) \} / \simeq , \]

where $\simeq $ is the equivalence relation on $\operatorname{Sing}_1(X)$ described by

\[ ( \sigma \simeq \sigma ' ) \Leftrightarrow ( \exists \tau \in \operatorname{Sing}_2(X) ) [ d^{2}_0(\tau ) = s^{0}_0(x) \text{ and } d^{2}_1(\tau ) = \sigma \text{ and } d^{2}_2(\tau ) = \sigma ' ]. \]

The datum of a $2$-simplex $\tau $ satisfying these conditions is equivalent to the datum of a continuous map $| \Delta ^2 | \rightarrow X$ with boundary behavior as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & x \ar [dr]^{ \underline{x} } \\ x \ar [ur]^{\sigma '} \ar [rr]^{\sigma } & & x; } \]

such a map can be identified with a homotopy between the paths determined by $\sigma $ and $\sigma '$.