📂Topological Data Analysis

Fundamental group in Algebraic Topology

Definition ¹

Given a topological space $X$ and a unit interval $I = [0,1]$,

1. For paths $f, g : I \to X$ in $X$, when $f (1) = g(0)$, the product or composition $f \cdot g$ of two paths is defined as: $$f \cdot g (s) := \begin{cases} f \left( 2s \right) & , \text{if } s \in [0, 1/2] \\ g \left( 2s - 1 \right) & , \text{if } s \in [1/2, 1] \end{cases}$$
2. For a path $f : I \to X$, a path defined as $\overline{f} : I \to X$ when $\overline{f} (s) := f (1-s)$ is called the inverse path of $f$.
3. For all $s_{1} , s_{2} \in I$, a path $c_{x_{0}}$ such that $c_{x_{0}} \left( s_{1} \right) = c_{x_{0}} \left( s_{2} \right) = x_{0}$, i.e., a constant function, is called a constant path.
4. When $f (0) = f(1) = x_{0} \in X$, i.e., the initial point and the terminal point of the path $f$ are the same, it is called a loop, and $x_{0} \in X$ is called the basepoint.
5. The set of all homotopy classes of loops $f$ based at $x_{0} \in X$ is denoted as $\pi_{1} \left( X , x_{0} \right)$. When defining a binary operation $\ast$ on homotopy classes of two loops $[f] , [g] \in \pi_{1} \left( X , x_{0} \right)$ as $$[f] \ast [g] := \left[ f \cdot g \right]$$ then the group $\pi_{1} \left( \left( X , x_{0} \right) \right)$ is called the fundamental group. Usually, $\ast$ is not even mentioned and is simply written as $[f] [g] = [f \cdot g]$.

Explanation

In mathematics, anything with “Fundamental” attached to it is of great importance. Initially, one may feel reluctant about how loops could lead to creating a group, but when considering their homotopy instead of just a set of loops, it becomes relatively easy to imagine its significance as it primarily concerns the properties of the topological space.

Product of Paths

The product of paths $f \cdot g$, as evident from the equation, follows path $f$ until $1/2 \le s \le 1$ and then follows path $g$ from then on. That such a product remains continuous is guaranteed by the pasting lemma.

Pasting Lemma: For a topological space $X,Y$, given two closed sets $A,B \subset X$ satisfying $A \cup B = X$, and two continuous functions $f : A \to Y$ and $g : B \to Y$ such that for all $x \in A \cap B$, $f(x) = g(x)$, then a function defined as $h$ is a continuous function. $$h(x) : = \begin{cases} f(x), & x \in A \\ g(x), & x \in B \end{cases}$$

Although the concept is not difficult, the terminology may feel awkward. Since paths are essentially functions, referring to their composition can be confusing due to potential confusion with composite functions, and the term product might also be confusing if there’s any operation in $X$ that could be considered multiplication. However, contrary to concerns, when studying, the operation $[f] [g] = [f \cdot g]$ within the homotopy classes is mainly mentioned, and describing $f \cdot g$ in words is rare.

Inverses and Identity Elements in Fundamental groups

An inverse path can be perceived as erasing the path taken in the product of paths. To intuitively understand this, consider $\overline{f}$ as simply being path $f$ but in the opposite direction.

Then, the product $f \cdot \overline{f}$ with $\overline{f}$ for any given path $f$ results in a loop since its starting and ending points are the same. Important to note is that the endpoint of $f$ and the starting point of $\overline{f}$, $x_{1}$, either precisely hits or simply remains static at $x_{0}$, which is a constant path $c_{x_{0}}$, or is homotopically equivalent. Since $\pi_{1} \left( X, x_{0} \right)$ is a set of homotopy classes, for all $f$, $$f \cdot \overline{f} \simeq c_{x_{0}}$$ would hold true. Seeing this, regardless of what $f$ is, its homotopy class $[f]$, through operation with the homotopy class of $\overline{f}$, $\left[ \overline{f} \right]$, always results in $\left[ c_{x_{0}} \right]$, and $c_{x_{0}}$ is evidently the identity element of $\pi_{1} \left( X , x_{0} \right)$ from its definition.

Simply Connected Spaces

From the discussions so far, one might wonder about the necessity of $x_{0}$ in fundamental groups.

For example, as shown in the picture above, it’s conceivable to think of a loop that ’erases’ the path before reaching a new point $x_{1} \in X$, making it seem indifferent to the choice of any basepoint $x_{0}$.

Change of Basepoint in Fundamental groups: Given a topological space $X$, let $h : I \to X$ be a path from $x_{0}$ to $x_{1}$. The function defined as $\beta_{h} : \pi_{1} \left( X , x_{1} \right) \to \pi_{0} \left( X_{1} , x_{0} \right)$ according to $\beta_{h} [f] := \left[ h \cdot f \cdot \overline{h} \right]$ is an isomorphism, known as the Change of Basepoint.

According to the above theorem, if $X$ is path-connected, then $\pi_{1} \left( X , x \right)$ is isomorphic regardless of the choice of basepoint $x$, hence, it is sometimes represented as $\pi_{1} \left( X \right)$ or more concisely as just $\pi_{1} X$.

Especially, if a space is path-connected and its fundamental group $\pi_{1} X$ is a trivial group, i.e., isomorphic to a finite group with only the identity element $e$ such that $\pi_{1} X \simeq \left\{ e \right\}$, then $X$ is considered a simply-connected space.

Conversely, this implies that the properties of a fundamental group can significantly vary with the choice of basepoint $x_{0}$, and even if it is path-connected, it is necessary to examine separately the algebraic properties it might ha