Fundamental group - Algebraic Structure and Functoriality

Functoriality and the Fundamental Group Introduction The fundamental group is not just an interesting invariant of topological spaces—it's also functorial. This means that continuous maps between spaces induce group homomorphisms between their fundamental groups in a way that respects the algebraic structure. Functoriality is what allows us to use algebraic methods to solve topological problems, and it's the bridge between topology and group theory. In this section, we'll explore how the fundamental group functor works, what tools we have for computing fundamental groups of complex spaces, and how covering spaces reveal the deeper structure of the fundamental group. Induced Homomorphisms The Basic Idea Suppose we have a continuous map $f:(X,x0)\to(Y,y0)$ that respects basepoints. This map doesn't just send points to points—it also sends loops to loops in a coherent way. If $\alpha$ is a loop in $X$ based at $x0$, then $f\circ\alpha$ is a loop in $Y$ based at $y0$ (since $f(x0)=y0$). More importantly, if two loops $\alpha$ and $\beta$ are homotopic in $X$, then $f\circ\alpha$ and $f\circ\beta$ are homotopic in $Y$. This is because homotopy respects continuous maps—if $H:\alpha\simeq\beta$ is a homotopy in $X$, then $f\circ H$ is a homotopy from $f\circ\alpha$ to $f\circ\beta$ in $Y$. Key point: This composition operation respects the group structure. If $\alpha$ and $\beta$ are loops, then the concatenation $\alpha\cdot\beta$ gets sent to $f\circ(\alpha\cdot\beta)=(f\circ\alpha)\cdot(f\circ\beta)$. This means $f$ induces a group homomorphism: $$f:\pi1(X,x0)\to\pi1(Y,y0)$$ defined by $f([\alpha])=[f\circ\alpha]$. Why This Matters The induced homomorphism $f$ is our tool for translating topological information into algebraic information. If two spaces have different fundamental groups, they cannot be homeomorphic (or even homotopy equivalent). Conversely, if $f$ is a particularly nice type of map, the properties of $f$ tell us something about $f$ itself. Homotopy Invariance The Fundamental Property Here's a remarkable fact: if two basepoint-preserving continuous maps $f$ and $g:(X,x0)\to(Y,y0)$ are homotopic (as basepoint-preserving maps), then they induce the same homomorphism on the fundamental group: $$f=g:\pi1(X,x0)\to\pi1(Y,y0)$$ Why? If $H:f\simeq g$ is a basepoint-preserving homotopy, and $[\alpha]\in\pi1(X,x0)$, then we can "slide" the loop $f\circ\alpha$ continuously into $g\circ\alpha$ using the homotopy. The homotopy class remains the same. The Consequence This property means that the fundamental group is homotopy invariant: two spaces that are homotopy equivalent have isomorphic fundamental groups. In particular, any space that is contractible (homotopy equivalent to a point) has trivial fundamental group. Products and Coproducts Products of Spaces For path-connected spaces $X$ and $Y$, the fundamental group of their product is the direct product of fundamental groups: $$\pi1(X\times Y, (x0,y0))\cong\pi1(X,x0)\times\pi1(Y,y0)$$ Intuition: A loop in a product space is a pair of loops, one in each coordinate. Homotopy in the product respects this structure, so loops in $X\times Y$ correspond exactly to pairs of loops in $X$ and $Y$. Example: The torus $S^1\times S^1$ has fundamental group $\mathbb{Z}\times\mathbb{Z}$, reflecting the two independent circular directions. Wedge Sums The wedge sum $X\vee Y$ is obtained by gluing two spaces at a single point. Its fundamental group is the free product of the individual fundamental groups: $$\pi1(X\vee Y)\cong\pi1(X)\pi1(Y)$$ Key difference from products: The free product is not commutative or abelian (in general), and it's strictly larger than the direct product. Intuitively, when you glue spaces at a point, you can form loops that weave between the two spaces without any constraint—they don't have to commute. Example: The wedge $S^1\vee S^1$ (two circles glued at a point) has fundamental group $\mathbb{Z}\mathbb{Z}$, the free group on two generators, which is non-abelian. The Seifert–van Kampen Theorem Statement and Intuition The Seifert–van Kampen Theorem is the fundamental tool for computing fundamental groups of spaces built from simpler pieces. Here's the basic setup: Suppose a space $X$ is the union of two open subsets $U$ and $V$ that are path-connected and overlap in a path-connected subset $U\cap V$. Then: $$\pi1(X)\cong\pi1(U){\pi1(U\cap V)}\pi1(V)$$ The right side is the free product with amalgamation, meaning we take the free product $\pi1(U)\pi1(V)$ but identify elements that come from the overlap. Why this works: A loop in $X$ can be decomposed into pieces that lie alternately in $U$ and $V$. The intersection $U\cap V$ is where these pieces must "connect," and this is captured algebraically by the free product with amalgamation. Special Case: Disjoint Union in the Intersection The most important special case occurs when $U\cap V$ is path-connected but has trivial fundamental group. Then: $$\pi1(X)\cong\pi1(U)\pi1(V)$$ This is a free product without amalgamation. Application: Fundamental Group of $S^2$ Let's see Seifert–van Kampen in action. We'll show that the 2-sphere $S^2$ has trivial fundamental group. Strategy: Decompose $S^2$ into two hemispheres, each contractible (and thus simply connected). Take $U$ and $V$ to be the two open hemispheres (with a slight overlap near the equator). Both $U$ and $V$ are contractible, so $\pi1(U)=\pi1(V)=\{e\}$. The intersection $U\cap V$ is a band around the equator, which is homotopy equivalent to $S^1$, but this doesn't matter for Seifert–van Kampen. By the theorem: $$\pi1(S^2)\cong\{e\}\{e\}=\{e\}$$ So $S^2$ is simply connected. Abelianization and First Homology The Connection There's a profound relationship between the fundamental group and homology. The abelianization of $\pi1(X)$—obtained by quotienting out the commutator subgroup $[\pi1(X),\pi1(X)]$—is naturally isomorphic to the first singular homology group: $$\pi1(X)^{\text{ab}} \cong H1(X;\mathbb{Z})$$ Why this is true: Homology is an abelian invariant (it's built from abelian groups), while the fundamental group is nonabelian in general. The first homology group captures the "abelian part" of the fundamental group—it forgets about how loops fail to commute, but remembers their 1-dimensional structure. Practical Consequence If you know $H1(X)$, you immediately know the abelianization of $\pi1(X)$. This often gives strong constraints on what $\pi1(X)$ can be. Covering Spaces Definition and Basic Properties A covering map is a continuous surjection $p:\tilde{X}\to X$ such that every point in $X$ has a neighborhood $U$ that is evenly covered by $p$: the preimage $p^{-1}(U)$ is a disjoint union of open sets, each mapping homeomorphically onto $U$. Intuition: A covering space is like a "multi-sheeted" copy of $X$, where $p$ folds the sheets down onto $X$. Simple example: The map $p:\mathbb{R}\to S^1$ given by $p(t)=e^{2\pi it}$ is a covering map. The real line "wraps around" the circle infinitely many times. Why Covering Spaces Matter Covering spaces are incredibly useful because: Properties that are hard to study on $X$ may be easier on the covering space $\tilde{X}$ The structure of covering spaces is completely encoded in the fundamental group (we'll see this next) Universal Covering Spaces Definition A universal covering space of $X$ is a covering map $p:\tilde{X}\to X$ where $\tilde{X}$ is simply connected (i.e., $\pi1(\tilde{X})=\{e\}$). Key fact: Every path-connected, locally path-connected space has a universal covering space, and it's unique up to homeomorphism. Example: The universal covering of $S^1$ is $\mathbb{R}$, with $p(t)=e^{2\pi it}$. The Fundamental Group from Covering Spaces Here's the bridge between covering spaces and the fundamental group: Theorem: If $p:\tilde{X}\to X$ is a universal covering, then the group of deck transformations (homeomorphisms $\phi:\tilde{X}\to\tilde{X}$ satisfying $p\circ\phi=p$) is isomorphic to $\pi1(X)$: $$\text{Deck}(\tilde{X}/X)\cong\pi1(X)$$ Intuition: Deck transformations permute the "sheets" of the covering space. For the universal cover, there's exactly one deck transformation for each homotopy class of loops in $X$. Concrete Example: The Circle For $p:\mathbb{R}\to S^1$ with $p(t)=e^{2\pi it}$: The deck transformations are translations $t\mapsto t+n$ for $n\in\mathbb{Z}$ These form a group isomorphic to $\mathbb{Z}$ We recover $\pi1(S^1)\cong\mathbb{Z}$ Classification of Covering Spaces The Main Result Universal covering spaces don't just exist for aesthetic reasons—they reveal the structure of $X$. Here's the fundamental classification theorem: For a path-connected, locally path-connected space $X$ with basepoint $x0$: there is a one-to-one correspondence between: Connected covering spaces of $X$ (up to isomorphism), and Subgroups of $\pi1(X,x0)$ Moreover, if $p:\tilde{X}\to X$ is a connected covering space and $\tilde{x}0\in p^{-1}(x0)$, then: $$\pi1(\tilde{X},\tilde{x}0)\cong H\subset\pi1(X,x0)$$ where $H$ is the subgroup corresponding to the covering. What This Means The fundamental group completely determines all covering spaces (at least the connected ones). If you know $\pi1(X)$, you know all possible coverings of $X$. Fibrations and the Long Exact Sequence What is a Fibration? A fibration is a map $p:E\to B$ satisfying a homotopy lifting property: given a homotopy in the base space $B$, you can always "lift" it to a homotopy in the total space $E$ (subject to consistency). Fibrations are more general than covering spaces and more flexible to work with. The Long Exact Sequence A fibration $p:E\to B$ with fiber $F$ (the typical preimage $p^{-1}(b0)$) induces a long exact sequence of homotopy groups: $$\cdots\to\pi2(B)\to\pi1(F)\to\pi1(E)\to\pi1(B)\to\pi0(F)\to\cdots$$ What this tells you: There's a precise algebraic relationship between the fundamental groups of $F$, $E$, and $B$. Given information about two of them, you can constrain the third. Example: Sphere Bundles One important application is computing the fundamental group of rotation groups. Using the fibration $SO(n)\to SO(n+1)\to S^n$, we can show that: $$\pi1(SO(n))\cong\mathbb{Z}2\quad\text{for }n\geq 3$$ This is a classical result: rotations in three or higher dimensions have only one non-trivial covering (the spinor group), reflecting the fact that rotating by $4\pi$ (not $2\pi$) gets you back to where you started. <extrainfo> Lie Groups and Universal Covers For connected, simply connected compact Lie groups $G$, any finite subgroup $\Gamma\subset G$ gives a quotient $G/\Gamma$ with $\pi1(G/\Gamma)\cong\Gamma$. This is how quotients of simply-connected Lie groups produce spaces with finite fundamental groups. </extrainfo> What Groups Can Be Fundamental Groups? Graphs and Free Groups Not every group can be the fundamental group of some space. There are significant restrictions. Key fact: The fundamental group of any graph (1-dimensional CW complex) is a free group. Moreover, every free group arises this way: if $Fn$ is the free group on $n$ generators, it equals $\pi1$ of a graph with $n$ independent loops. Conversely, only free groups can be realized as the fundamental group of a graph. This is because you can build any loop in a graph by traversing edges, and you can go backwards on any edge (unlike in a manifold), so there are no relations—everything is free. General Realizability The question of which groups can be realized as $\pi1(X)$ for some space $X$ is more subtle: Every group can be realized as the fundamental group of some space (we can build it explicitly) But realizing a group as the fundamental group of a "nice" space (like a smooth manifold) is highly restrictive For example, the fundamental group of a smooth manifold must be a group that satisfies certain finiteness properties. Not every group satisfies these. Key Computational Strategies Building Spaces from Simpler Pieces When faced with a complex space, look for ways to decompose it: Products: If $X=A\times B$ and both are path-connected, then $\pi1(X)\cong\pi1(A)\times\pi1(B)$ Wedge sums: If $X=A\vee B$, then $\pi1(X)\cong\pi1(A)\pi1(B)$ (more generally, a free product with amalgamation) Seifert–van Kampen: For arbitrary unions of path-connected open sets with path-connected intersection Using Covering Spaces If a space has a known covering space: Use the long exact sequence of the fibration Use the correspondence between subgroups and coverings Sometimes it's easier to compute $\pi1$ of the covering and work backwards Abelianization If you only need to know the abelian quotient, compute $H1$ using homology tools—it's often much faster than computing $\pi1$ directly.