Fundamental group Study Guide

📖 Core Concepts Loop & Base Point – A continuous map $\ell:[0,1]\to X$ with $\ell(0)=\ell(1)=x0$. Homotopy of Loops – A deformation $H:[0,1]\times[0,1]\to X$ keeping the base point fixed; $\ell0\simeq\ell1$ means they lie in the same class. Fundamental Group $\pi1(X,x0)$ – Set of homotopy classes $[\ell]$ with product given by concatenation $$ (fg)(s)=\begin{cases} f(2s),&0\le s\le\tfrac12,\\[4pt] g(2s-1),&\tfrac12\le s\le1 . \end{cases} $$ Group Structure – Identity = constant loop; inverse = reverse loop $f^{-1}(t)=f(1-t)$. Base‑Point Dependence – In a path‑connected space any two base points give isomorphic groups (change‑of‑base‑point isomorphism, unique up to inner automorphism). Simply Connected – $\pi1(X)=\{e\}$; equivalently every loop contracts to the constant loop. Key Examples $\mathbb R^n$ or any convex set → trivial $\pi1$. $S^1$ → $\pi1(S^1)\cong\mathbb Z$ (winding number). Figure‑eight → free group $F2$. Torus $T^2$ → $\pi1(T^2)\cong\mathbb Z\times\mathbb Z$. Connected graph → free group of rank $E-V+1$. Functoriality – A base‑point‑preserving map $f:(X,x0)\to(Y,y0)$ induces a homomorphism $f:\pi1(X)\to\pi1(Y)$. Van Kampen Theorem – Gluing two path‑connected opens $U,V$ (with path‑connected intersection) yields $$\pi1(X)\cong \pi1(U){\pi1(U\cap V)}\pi1(V).$$ Covering Spaces – $p:\widetilde X\to X$ with evenly covered neighborhoods; the deck‑transformation group $\operatorname{Deck}(p)\cong\pi1(X)$ when $\widetilde X$ is universal (simply connected). --- 📌 Must Remember Definition: $\pi1(X,x0)=\{[\ell]\mid \ell\text{ loop at }x0\}$, product = concatenation. Identity: constant loop $c(t)=x0$. Inverse: $\ell^{-1}(t)=\ell(1-t)$. Associativity: holds up to homotopy; the group law is well‑defined on classes. Base‑point change: If $\gamma$ is a path $x0\to x1$, then $$\Phi\gamma([\ell])=[\gamma\ell\gamma^{-1}] : \pi1(X,x0)\to\pi1(X,x1).$$ Trivial groups: $\pi1(\mathbb R^n)=0$, $\pi1(\text{convex set})=0$, $\pi1(S^n)=0$ for $n\ge2$. Circle: $\pi1(S^1)=\mathbb Z$, generator = once‑around loop. Graph rank: $r=E-V+1$; $\pi1(\text{graph})\cong Fr$. Surface of genus $g$: presentation $$\langle a1,b1,\dots,ag,bg\mid\prod{i=1}^g[ai,bi]=1\rangle.$$ Product rule: $\pi1(X\times Y)\cong\pi1(X)\times\pi1(Y)$. Wedge rule: $\pi1(X\vee Y)\cong\pi1(X)\pi1(Y)$. Universal cover of $S^1$: $p:\mathbb R\to S^1$, $p(t)=e^{2\pi i t}$, deck group $\mathbb Z$. Abelianization: $\pi1(X)^{\text{ab}}\cong H1(X)$. --- 🔄 Key Processes Compute $\pi1$ of a Graph Count vertices $V$ and edges $E$. Form a spanning tree $T$ (uses $V-1$ edges). Remaining $E-(V-1)$ edges → generators. No relations → free group $F{E-V+1}$. Apply Van Kampen Choose open sets $U$, $V$ covering $X$ with path‑connected overlap. Write down $\pi1(U)$, $\pi1(V)$, $\pi1(U\cap V)$. Form the free product of $\pi1(U)$ and $\pi1(V)$ and impose the identifications coming from $\pi1(U\cap V)$. Change Base Point Pick a path $\gamma$ from $x0$ to $x1$. Conjugate every loop class by $\gamma$ as in the formula above. Construct a Universal Cover (when feasible) Start with a simply connected space $\widetilde X$ that locally looks like $X$. Define $p:\widetilde X\to X$ by “folding” $\widetilde X$ onto $X$. Verify that deck transformations act freely and transitively on fibers; the group of deck transformations is $\pi1(X)$. Abelianize $\pi1$ Quotient $\pi1$ by its commutator subgroup $[\pi1,\pi1]$. Result equals the first homology $H1(X)$. --- 🔍 Key Comparisons Simply connected vs. Trivial $\pi1$ – Same: a space is simply connected iff its fundamental group is the trivial group. Free group vs. Abelian group – Free: non‑abelian unless rank ≤ 1; Abelian: all generators commute (e.g., $\mathbb Z^n$). Base‑point‑dependent vs. Base‑point‑independent – Dependent: raw definition uses a point; Independent: for path‑connected spaces, groups at different points are isomorphic (up to inner automorphism). Concatenation associativity – Strict: in the group of classes; Only up to homotopy: on the level of actual loops. Graph fundamental groups vs. Surface groups – Graphs: always free; Surfaces: have a single relation $\prod [ai,bi]=1$. --- ⚠️ Common Misunderstandings “Concatenation is associative” – Only after passing to homotopy classes; raw loops need a re‑parametrization. “Trivial $\pi1$ ⇒ contractible” – False; $S^2$ is simply connected but not contractible. “All spaces have abelian $\pi1$” – Only spaces homotopy‑equivalent to products of circles (or topological groups) guarantee abelian $\pi1$. “Base point never matters” – In non‑path‑connected spaces the groups can be unrelated; even in path‑connected spaces the isomorphism depends on a chosen path (inner automorphism). “Covering space classification = subgroups of any group” – Only subgroups of $\pi1(X)$ correspond to connected covering spaces. --- 🧠 Mental Models / Intuition Rubber‑band picture: Loops are stretchy bands; homotopy = sliding the band without cutting; $\pi1$ records distinct ways the band can be slipped around holes. Winding number for $S^1$: Count how many times the rubber band wraps; positive = one direction, negative = opposite. Spanning tree: Think of the tree as a “skeleton” that can be traversed without creating a loop; each extra edge adds a new independent loop (a new generator). Covering space: Imagine an infinite staircase (the universal cover of $S^1$); each step corresponds to a loop class; moving one step = applying the generator $1\in\mathbb Z$. --- 🚩 Exceptions & Edge Cases Non‑path‑connected spaces: $\pi1$ is defined only after choosing a component; different components can have unrelated groups. Topological groups: $\pi1$ is always abelian, regardless of the space’s other features. Changing base point in non‑simply‑connected spaces: The isomorphism may be non‑trivial (inner automorphism), affecting how specific generators are identified. Graphs with multiple components: Each component has its own free group; the overall $\pi1$ is the free product of the component groups. --- 📍 When to Use Which Van Kampen → When the space can be split into two (or more) overlapping open subsets with simpler fundamental groups (e.g., wedge of circles, surfaces cut into disks). Spanning‑tree method → Best for 1‑dimensional CW complexes (graphs, simplicial 1‑skeleta). Product rule → For product spaces like $S^1\times S^1$ (torus) or $X\times Y$ where both are path‑connected. Wedge rule → For spaces formed by gluing at a single point (e.g., figure‑eight, bouquet of circles). Universal covering → When you need to compute $\pi1$ indirectly (e.g., $\pi1(S^1)$ from $\mathbb R$) or to classify covering spaces. Abelianization → When the question asks for $H1(X)$ or when only the commutative information of $\pi1$ matters. --- 👀 Patterns to Recognize “One extra edge → one generator” – In any graph, each edge not in a spanning tree gives a loop generator. “Wedge = free product” – Whenever spaces meet at a single point, their $\pi1$’s combine via free product. “Product = direct product” – For Cartesian products of path‑connected spaces. “Covering deck group = $\pi1$” – Spot a covering map; the symmetry group of the covering equals the fundamental group of the base. “Relations come from 2‑cells (triangles, disks)” – In a CW complex, each 2‑cell adds a relation; in graphs (no 2‑cells) there are none, so the group is free. --- 🗂️ Exam Traps Choosing the wrong base point – Forgetting to state that the answer is up to isomorphism; some problems expect the explicit change‑of‑base‑point map. Assuming $\pi1(T^2)=\mathbb Z$ – The torus has two independent loops, so $\pi1(T^2)=\mathbb Z\times\mathbb Z$. Treating concatenation as strictly associative – A distractor may present a “proof” that fails because re‑parametrization is ignored. Mixing up homotopy of maps vs. homotopy of loops – The former need not fix the base point; only the latter define $\pi1$. Believing any group can be realized by a graph – Only free groups arise from graphs; a non‑free group (e.g., $\mathbb Z\times\mathbb Z$) cannot be a graph’s $\pi1$. Overlooking inner automorphisms in base‑point change – Two isomorphisms may differ by conjugation; answer choices that ignore this subtlety are wrong. Confusing abelianization with the original group – $F2$ abelianizes to $\mathbb Z^2$, but $F2$ itself is non‑abelian. ---