nLab homotopy relative boundary

For paths

Definition

Let $X$ be a topological space and let

$$\gamma_1, \gamma_2 \;\colon\; [0,1] \longrightarrow X$$

be two paths in $X$, i.e. two continuous functions from the closed interval to $X$, such that their endpoints agree:

$$\gamma_1(0) = \gamma_2(0) \phantom{AAAA} \gamma_1(1) = \gamma_2(1) \,.$$

Then a homotopy relative boundary from $\gamma_1$ to $\gamma_2$ is a homotopy (def. )

$$\eta \;\colon\; \gamma_1 \Rightarrow \gamma_2$$

such that it does not move the endpoints:

$$\eta(0,-) = const_{\gamma_1(0)} = const_{\gamma_2(0)} \phantom{AAAAAA} \eta(1,-) = const_{\gamma_1(1)} = const_{\gamma_2(1)} \,.$$

Proposition

Let $X$ be a topological space and let $x, y \in X$ be two points. Write

$$P_{x,y} X$$

for the set of paths $\gamma$ in $X$ with $\gamma(0) = x$ and $\gamma(1) = y$.

Then homotopy relative boundary (def. ) is an equivalence relation on $P_{x,y}X$.

The corresponding set of equivalence classes is denoted

$$Hom_{\Pi_1(X)}(x,y) \;\coloneqq\; (P_{x,y}X)/\sim \,.$$

The operation of path concatenation descends to these equivalence classes, so that for all $x,y, z \in X$ there is a function

$$Hom_{\Pi_1(X)}(x,y) \times Hom_{\Pi_1(X)}(y,z) \longrightarrow Hom_{\Pi_1(X)}(x,z) \,.$$

Moreover, this composition operation is associative in the evident sense.

Set of points of $X$ together with the set $Hom_{\Pi_1(X)}(x,y)$ for all points of points and equipped with this composition operation is called the fundamental groupoid of $X$, denoted

$$\Pi_1(X) \,.$$

If we pick a single point $x \in X$, then one writes

$$\pi_1(X,x) \;\coloneqq\; Hom_{\Pi_1(X)}(x,x) \,.$$

Under the above composition this is a group, and as such is called the fundamental group of $X$ at $x$.