4.2 The fundamental group

The fundamental group is a powerful tool in algebraic topology, capturing the essence of a space's "loopiness." It measures how loops in a space can be continuously deformed into each other, revealing crucial information about the space's structure and holes.

By studying the fundamental group, we gain insights into a space's topology that go beyond simple connectivity. This concept forms the foundation for understanding higher-dimensional topological features and is essential for classifying spaces up to homotopy equivalence.

Fundamental Group of a Space

Definition and Notation

• Define the fundamental group $\pi_{1}(X, x_{0})$ of a topological space $X$ at a basepoint $x_{0}$ as the set of homotopy equivalence classes of loops based at $x_{0}$
• Denote a loop in $X$ based at $x_{0}$ as a continuous function $f: [0, 1] \rightarrow X$ such that $f(0) = f(1) = x_{0}$
• Consider two loops $f$ and $g$ homotopic if there exists a continuous function $H: [0, 1] \times [0, 1] \rightarrow X$ such that:
  • $H(s, 0) = f(s)$ and $H(s, 1) = g(s)$ for all $s \in [0, 1]$
  • $H(0, t) = H(1, t) = x_{0}$ for all $t \in [0, 1]$

Group Structure

• Define the group operation in $\pi_{1}(X, x_{0})$ by concatenation of loops
  • The inverse of a loop is the same loop traversed in the opposite direction
• Identify the identity element in $\pi_{1}(X, x_{0})$ as the constant loop at $x_{0}$
• Recognize that the fundamental group is a homotopy invariant
  • If $X$ and $Y$ are homotopy equivalent spaces, then $\pi_{1}(X, x_{0}) \cong \pi_{1}(Y, y_{0})$ for any choice of basepoints $x_{0}$ and $y_{0}$
• Understand that if $X$ is a path-connected space, the fundamental groups at different basepoints are isomorphic
  • The isomorphism is induced by conjugation with paths connecting the basepoints

Geometric Interpretation of the Fundamental Group

Loops and Holes

• Interpret the fundamental group as capturing the notion of loops in a space that cannot be continuously deformed to a point within the space
• Recognize that elements of the fundamental group represent different ways to loop around holes or obstacles in the space
  • Non-trivial elements correspond to loops that encircle holes or obstacles and cannot be contracted to a point
• Conclude that if $\pi_{1}(X, x_{0})$ is trivial (consists only of the identity element), every loop in $X$ based at $x_{0}$ can be continuously shrunk to the point $x_{0}$ without leaving the space

Covering Spaces

• Relate the fundamental group of a covering space to the fundamental group of the base space by the lifting criterion
  • A loop in the base space lifts to a loop in the covering space if and only if it represents an element of the fundamental group that lies in the image of the induced homomorphism from the covering space
• Understand that the fundamental group of a covering space is a subgroup of the fundamental group of the base space
  • The index of this subgroup is equal to the number of sheets in the covering space

Computing the Fundamental Group

Simple Spaces

• Recognize that the fundamental group of a convex subset of Euclidean space is trivial
  • Any loop can be continuously shrunk to a point within the space
• Compute the fundamental group of a circle $S^{1}$ as isomorphic to the integers under addition, $\pi_{1}(S^{1}) \cong (\mathbb{Z}, +)$
  • Each integer represents the number of times a loop winds around the circle (winding number)
• Determine that the fundamental group of a punctured plane $\mathbb{R}^{2} \setminus \{0\}$ is also isomorphic to $(\mathbb{Z}, +)$
  • Loops can wind around the removed point

Product Spaces and Wedge Sums

• Prove that the fundamental group of a product space $X \times Y$ is isomorphic to the direct product of the fundamental groups of $X$ and $Y$
  • $\pi_{1}(X \times Y, (x_{0}, y_{0})) \cong \pi_{1}(X, x_{0}) \times \pi_{1}(Y, y_{0})$
• Calculate the fundamental group of a figure-eight space (two circles joined at a single point) as the free group on two generators
  • The generators correspond to loops around each circle
• Compute the fundamental group of a torus as isomorphic to $\mathbb{Z} \times \mathbb{Z}$
  • The generators are loops that wind around the two independent directions of the torus (meridian and longitude)

Properties of the Fundamental Group

Homotopy Invariance

• Understand that the fundamental group is a homotopy invariant
  • If two spaces $X$ and $Y$ are homotopy equivalent, then their fundamental groups $\pi_{1}(X, x_{0})$ and $\pi_{1}(Y, y_{0})$ are isomorphic for any choice of basepoints $x_{0}$ and $y_{0}$
• Apply homotopy invariance to simplify the computation of fundamental groups
  • If a space is homotopy equivalent to a simpler space with a known fundamental group, the original space will have the same fundamental group (deformation retract)

Basepoint Independence

• Recognize that for path-connected spaces, the fundamental groups at different basepoints are isomorphic
• Understand that the isomorphism between fundamental groups at different basepoints is induced by conjugation with paths connecting the basepoints
  • If $\gamma$ is a path from $x_{0}$ to $x_{1}$ in $X$, then the isomorphism $\pi_{1}(X, x_{0}) \to \pi_{1}(X, x_{1})$ is given by $[\alpha] \mapsto [\gamma^{-1} * \alpha * \gamma]$, where $*$ denotes path concatenation
• Apply basepoint independence to simplify the computation of fundamental groups
  • Choose a convenient basepoint for calculations, knowing that the resulting fundamental group will be isomorphic to the fundamental group at any other basepoint in the same path component