Fundamental group - Examples and Computations

Computing Fundamental Groups: Methods and Examples Introduction The fundamental group is a powerful invariant that measures the "hole structure" of a space using loops. While computing fundamental groups from first principles can be difficult, we can leverage several concrete examples and computational techniques. This guide covers the most important methods: recognizing spaces with known fundamental groups, using edge-path groups in simplicial complexes, analyzing graphs, and applying van Kampen's theorem. Together, these tools allow us to compute the fundamental groups of a wide variety of spaces. Concrete Examples: Recognizing Fundamental Groups Euclidean Space and Convex Subsets Any convex subset of $\mathbb{R}^n$ (such as an open ball or the entire space itself) is simply connected. This means every loop can be continuously deformed to a point—intuitively, there's nothing in the space to "wind around." Therefore, the fundamental group of any convex space is the trivial group containing only the identity element. This is our most basic example and serves as a baseline for comparison. Simply Connected Spaces A space whose fundamental group is trivial (containing only the identity) is called simply connected. Beyond Euclidean space and its convex subsets, there are many other simply connected spaces. Crucially, the $2$-sphere $S^2$ and all higher-dimensional spheres $S^n$ for $n \geq 2$ are simply connected. This is a key fact to remember: while $S^1$ (the circle) has a non-trivial fundamental group, higher-dimensional spheres do not. Intuitively, in higher dimensions there is "room to move around" loops, allowing them to contract to a point even when they might appear to wind around something. The Circle: $\pi1(S^1) \cong \mathbb{Z}$ The circle $S^1$ is our first example of a space with a non-trivial fundamental group. Unlike higher spheres, the circle cannot be contracted to a point. The fundamental group is determined by winding number: every loop on $S^1$ can be classified by how many times it winds around the circle. A loop that winds counterclockwise $n$ times has winding number $n$; a loop that winds clockwise $m$ times has winding number $-m$. If a loop doesn't wind around at all, its winding number is $0$. The key insight is that concatenation of loops adds their winding numbers. If loop $\alpha$ winds around $m$ times and loop $\beta$ winds around $n$ times, then their concatenation winds around $m + n$ times. This addition operation corresponds exactly to the group operation on $\mathbb{Z}$. Therefore: $$\pi1(S^1) \cong \mathbb{Z}$$ This is one of the most important concrete examples in algebraic topology. The Figure Eight: $\pi1 \cong F2$ (Free Group on Two Generators) The figure eight is formed by joining two circles at a single point. Choose this common point as your base point. Any loop on the figure eight can be described by specifying which circle you traverse and in which direction—and you can do this multiple times. For example, you might go around the first circle once, then the second circle twice, then back around the first circle in the opposite direction. We can encode such loops using two symbols, $a$ and $b$, representing the two circles. Then any loop becomes a "word" like $aba^{-1}b^2$ (around first circle, around second circle, around first circle backward, around second circle twice). The fundamental group is the free group on two generators, denoted $F2$. The term "free" means there are no relations among the generators—you can form any word you want, and two words represent the same loop only if you can transform one into the other by moving around the base point, which doesn't actually impose any simplifications. Key distinction: The free group $F2$ is non-abelian. This means $ab \neq ba$ in general. Geometrically, this makes sense: going around circle $a$ then circle $b$ traces a different path than going around $b$ then $a$. Graphs: Free Groups with Rank Equal to Cyclomatic Complexity For a connected graph with $E$ edges and $V$ vertices, the fundamental group is a free group whose rank is: $$\text{rank} = E - V + 1$$ Why this formula? A spanning tree of the graph uses exactly $V - 1$ edges and is simply connected. The remaining $E - (V-1) = E - V + 1$ edges each create one independent cycle, and each cycle becomes a generator of the fundamental group. How to identify generators: Choose a spanning tree. Each edge not in the tree, together with a path in the tree, forms a loop. These loops form a basis for $\pi1(\text{graph})$, which is therefore: $$\pi1(\text{graph}) \cong F{E-V+1}$$ Special cases: If the graph is a tree (no cycles), then $E = V - 1$, so rank $= 0$, and $\pi1$ is trivial. If the graph is a single cycle, then $V = E$, so rank $= 1$, and $\pi1 \cong \mathbb{Z}$ (exactly like the circle). Orientable Surfaces of Genus $g$ An orientable surface of genus $g$ is a surface with $g$ "holes" (or "handles"). The torus is genus $1$; a sphere is genus $0$. The fundamental group of an orientable surface of genus $g$ has the presentation: $$\pi1 \cong \langle a1,b1,\dots ,ag,bg \mid \prod{i=1}^g [ai,bi]=1\rangle$$ where $[ai,bi] = ai bi ai^{-1} bi^{-1}$ is the commutator of $ai$ and $bi$. For the torus ($g = 1$), this simplifies to: $$\pi1(\text{torus}) \cong \langle a, b \mid [a,b] = 1 \rangle \cong \mathbb{Z} \times \mathbb{Z}$$ The fact that $[a,b] = 1$ means the two generators commute, so the group is abelian. Geometrically, the two generators correspond to loops going around the two holes of the torus in orthogonal directions. For higher genus surfaces, the group remains non-abelian (except through the single relation), and is typically harder to understand directly. <extrainfo> Topological Groups For any topological group $G$ (a group equipped with a topology making the group operations continuous), the fundamental group based at the identity element is always abelian. This is a useful fact but somewhat specialized. It tells us, for instance, that the fundamental group of any Lie group is abelian. </extrainfo> Edge-Path Groups in Simplicial Complexes Definition and Equivalence In a connected simplicial complex $K$, an edge-path is a sequence of vertices $v0, v1, \dots, vn$ where each consecutive pair $(vi, v{i+1})$ forms an edge in $K$. Two edge-paths are edge-equivalent if one can be obtained from the other by a sequence of local moves. Specifically, you can replace any edge sequence $vi \to vj \to vk$ with $vi \to v\ell \to vk$ whenever $vi, vj, vk, v\ell$ form the three vertices of a triangle (2-simplex) in $K$. This replacement can go either direction. The Edge-Path Group Fix a base vertex $v0$. The edge-path group consists of all edge-loops (edge-paths starting and ending at $v0$) modulo edge-equivalence, with concatenation as the group operation. The crucial theorem is that the edge-path group is naturally isomorphic to the fundamental group $\pi1(|K|, v0)$ of the geometric realization of $K$. This gives us a combinatorial way to compute the fundamental group: we only need to work with vertices and edges of the simplicial complex, rather than considering all possible loops in the underlying space. Computation Using a Spanning Tree To compute the edge-path group explicitly: Choose a maximal spanning tree $T$ in the 1-skeleton (the graph made up of vertices and edges) of $K$. Generators: Each oriented edge not in $T$ becomes a generator of the edge-path group. Relations: Each triangle (2-simplex) in $K$ gives a relation. The three edges of the triangle satisfy a relation when expressed in terms of the generators. This method systematically converts the topological problem into an algebraic one: determining the presentation of a group given by generators and relations. Fundamental Groups of Connected Graphs Graphs as CW Complexes A connected graph can be viewed as a 1-dimensional CW complex: vertices are 0-cells and edges are 1-cells. Unlike simplicial complexes, graphs have no higher-dimensional cells. The fundamental group of a connected graph is always a free group, with rank equal to the number of independent cycles in the graph. Computing the Rank For a graph with $V$ vertices and $E$ edges, the rank of the fundamental group is: $$r = E - V + 1$$ This equals the size of any maximum set of edge-disjoint cycles. Equivalently, it's the number of edges you must remove to make the graph acyclic (into a tree). Identifying Generators Choose any spanning tree of the graph. Each of the $r = E - V + 1$ edges not in the spanning tree corresponds to a generator of $\pi1$. Specifically, each such edge, together with the unique path in the spanning tree connecting its endpoints, forms a loop. These $r$ loops generate the free group $\pi1(\text{graph}) \cong Fr$. Example: The figure eight has $V = 1$ vertex (two circles touching at one point) and $E = 2$ edges. So $r = 2 - 1 + 1 = 2$, giving $\pi1 \cong F2$, as we saw earlier. Non-Abelian Structure Unlike the circle, most graphs have non-abelian fundamental groups. The only exceptions are: Trees: No cycles, so $r = 0$ and $\pi1$ is trivial. Single cycles: One cycle, so $r = 1$ and $\pi1 \cong \mathbb{Z}$, which is abelian. Any graph with two or more independent cycles has a non-abelian fundamental group, since free groups on two or more generators are non-abelian. Van Kampen's Theorem Statement Van Kampen's theorem is a powerful tool for computing fundamental groups by decomposing a space into simpler pieces. Theorem: If $X$ is a path-connected space that is the union of two path-connected open sets $U$ and $V$ such that $U \cap V$ is also path-connected, then: $$\pi1(X) = \pi1(U) {\pi1(U \cap V)} \pi1(V)$$ This is the free product of $\pi1(U)$ and $\pi1(V)$ amalgamated over $\pi1(U \cap V)$. What this means: We can compute $\pi1(X)$ by taking the free product of the fundamental groups of the two pieces, but then imposing relations that identify elements from $\pi1(U)$ and $\pi1(V)$ that correspond to the same element in $\pi1(U \cap V)$. Practical Application to Graphs To compute $\pi1$ of a graph using van Kampen: Decompose the graph into two or more subgraphs whose intersections are simple (ideally trees or single edges, which are path-connected). Compute $\pi1$ of each piece. Since fundamental groups of graphs are free, this is manageable. Apply van Kampen: if the intersection is a tree, it has trivial fundamental group, so the intersection term vanishes and you get the free product $\pi1(U) \pi1(V)$. Example: The Figure Eight Revisited Consider the figure eight as the union of two disks (neighborhoods of the two circles) $U$ and $V$ that overlap in a small path-connected region around the touching point. $\pi1(U) \cong \mathbb{Z}$ (from the first circle) $\pi1(V) \cong \mathbb{Z}$ (from the second circle) $U \cap V$ is path-connected but contractible, so $\pi1(U \cap V)$ is trivial By van Kampen: $$\pi1(\text{figure eight}) = \mathbb{Z} \mathbb{Z} = F2$$ The free product of two copies of $\mathbb{Z}$ is the free group on two generators, confirming our earlier result.