Geometric Topology 16

Paths in Topological Spaces

From now on $X$ will denote a topological space. Here are some examples:

• Euclidean space $\mathbb{R}^n$, including the real line, the plane and ordinary space ($n=1,2,3$). The Argand plane $\mathbb{C}$, which unlike $\mathbb{R}^2$ is 'pre-oriented'.
• The $n$-dimensional sphere $S^n = \{\underline{v} \in \mathbb{R}^{n+1} : \|\underline{v}\| = 1\}$, which is compact. Basic topology asserts that $S^n$ minus a point is homeomorphic to $\mathbb{R}^n$.
• The $n$-dimensional torus $T^n$, which is homeomorphic to the product $S^1 \times \cdots \times S^1$ of $n$ circles, and to the quotient (group and top. space) $\mathbb{R}^n /\mathbb{Z}^n$. The $3$-torus $T^3$ is obtained from a solid cube by identifying opposite sides.
• Any graph in $\mathbb{R}^2$ or $\mathbb{R}^3$, including the figure-eight $"\infty" \approx q(\partial \mathscr{P})$ arising from the boundary code $aba^{-1}b^{-1}$ of the square $\mathscr{P}$ representing $T$.
• Any surface, in particular an orientable $\mathscr{M}_{Y,g,r}$ or a non-orientable $\mathscr{M}_{N,h,r}$.
• More general quotients of polygons like $'xxx'$ that are not locally Euclidean.
• The complement of a knot (this $3$-manifold is an open subset of $\mathbb{R}^3$ or $S^3$).

Joining Paths

Let $X$ be a topological space, $x_0,x_1 \in X$. Let $I = [0,1]$.

Definition: A path from $x_0$ to $x_1$ is a continuous map $\alpha : I \rightarrow X$ such that $\alpha(0) = x_0$ and $\alpha(1) = x_1$. There is no requirement for $\alpha$ to be injective. If $x_0 = x_1$ then it is called a loop based at $x_0$. Once $x_0$ is fixed, $s \mapsto x_0 \ (\forall s)$ defines the constant loop $\varepsilon$ at $x_0$.

Assume that $X$ is path-connected, i.e. any two points of $X$ can be joined by a path. This implies that $X$ is connected; these two properties are equivalent for surfaces.

If $\alpha$ is a path from $x_0$ to $x_1$, and $\beta$ is a path from $x_1$ to $x_2$, then $\alpha \beta$ will denote the concatenated path, defined by traversing $\alpha$ then $\beta$ both at double speed:

This is continuous: its image has no 'gap'.

We can define $\alpha \beta \gamma$ as $(\alpha \beta) \gamma$, or else divide into thirds (it won't really matter, we will see later). We also define a 'backwards path' $\alpha^{-1}$ by $\alpha^{-1}(s) = \alpha(1-s)$.

Homotopy of Paths

Definition: Fix $x_0,x_1 \in X$. Two paths $\alpha,\beta$ from $x_0$ to $x_1$ are path-homotopic if there exists a continuous mapping $H: I^2 \rightarrow X$ such that:

We will write $\alpha \approxeq \beta$ or $\alpha \approxeq_{H} \beta$.

The map $H$ represents the deformation of the paths fixing their endpoints, and it helps to write $H(s,t) = H_t(s)$ so $H_0 = \alpha$ and $H_1 = \beta$.

Ordinary homotopy $(\alpha \simeq \beta)$ would only insist on the left-hand equations, and then any path starting from $x_0$ would be homotopic to $\varepsilon$ by setting $H(s,t) = \alpha((1-t)s)$.

Lemma 0: $\approxeq$ is an equivalence relation on the set of all paths from $x_0$ to $x_1$. We will denote such an equivalence class by $[\alpha]$.

Proof: To see $\alpha \approxeq \alpha$, take $F(s,t) = \alpha(s) \ \forall t.$ To see that $\alpha \approxeq_F \beta$ implies $\beta \approxeq \alpha$, take:

To see that $\alpha \approxeq_F \beta$ and $\beta \approxeq_G \gamma$ implies that $\alpha \approxeq \gamma$, take:

so this time speed is doubled vertically. To see that $H$ is continuous, it suffices to show that if $C$ is a closed subset of $X$ then $H^{-1}(C)$ is a closed subset of $I$. Let $A = I \times [0,\frac{1}{2}]$ and $B = I \times [\frac{1}{2},1]$, noting that the squashed versions $\tilde{F},\tilde{G}$ agree on $A \cap B$ (where $t = \frac{1}{2}$). Then:

is the union of two closed subsets of $I$. $\blacksquare$

Loops in a Graph

Any graph can be regarded as a subset of $\mathbb{R}^3$, so that edges only meet at vertices. The diagram shows that $K_4$ (the complete tetrahedral graph with $4$ vertices) is planar, since it can be embedded as a subset of $\mathbb{R}^2$. Each face defines a loop based at $x_0$ by concatenation:

Note that $\delta$ resembles insertion of a cuff at the vertex $x_0$ of a polygon. Moreover $\alpha a = bda^{-1}a \approxeq bd$. This path-homotopy will be proved in the following section, but matches our treatment of boundary codes.

Using similar equivalences:

We shall show that this path-homotopy of loops can be interpreted as a relation in the so-called fundamental group. The latter is generated by the path-homotopy classes of loops based at a fixed point of a topological space (like $x_0$ in $K_4$).