Homotopy & the Fundamental Group
Homotopy & the Fundamental Group
Two continuous maps $f,g:X\to Y$ are homotopic if one can be continuously deformed into the other. The fundamental group $\pi_1(X,x_0)$ classifies loops at a basepoint up to homotopy — it captures the 1-dimensional 'holes' in $X$. Simply connected spaces (trivial fundamental group, like $\mathbb{R}^n$ and $S^n$ for $n\geq 2$) have no 1D holes. The fundamental group distinguishes $S^1$ (with $\pi_1=\mathbb{Z}$) from $S^2$ (with $\pi_1=0$).
Homotopy & Homotopy Equivalence
Maps $f,g:X\to Y$ are homotopic ($f\simeq g$) if $\exists H:X\times[0,1]\to Y$ continuous with $H(x,0)=f(x)$, $H(x,1)=g(x)$. Spaces $X\sim Y$ are homotopy equivalent ($X\simeq Y$) if $\exists f:X\to Y$, $g:Y\to X$ with $g\circ f\simeq\text{id}_X$ and $f\circ g\simeq\text{id}_Y$. Homotopy equivalence is weaker than homeomorphism: $\mathbb{R}^n\simeq\{\text{pt}\}$ (contractible); $S^1\simeq$ Möbius band; $\mathbb{R}^2\setminus\{0\}\simeq S^1$.
Fundamental Group $\pi_1$
Fix basepoint $x_0\in X$. A loop at $x_0$: continuous $\gamma:[0,1]\to X$ with $\gamma(0)=\gamma(1)=x_0$. Homotopy classes of loops $[\gamma]$ with basepoint-preserving homotopy form the fundamental group $\pi_1(X,x_0)$ under path concatenation $[\alpha][\beta]=[\alpha*\beta]$. Key computations: $\pi_1(\mathbb{R}^n)=0$ (simply connected); $\pi_1(S^1)\cong\mathbb{Z}$ (winding number); $\pi_1(S^n)=0$ for $n\geq 2$; $\pi_1(T^2)\cong\mathbb{Z}\times\mathbb{Z}$ (torus has two independent loops).
Example 1
Show $\pi_1(S^1)\cong\mathbb{Z}$.
Solution: A loop in $S^1=\{e^{i\theta}\}$ at basepoint $1$ wraps around an integer number of times — the winding number. Formally, lift $\gamma:[0,1]\to S^1$ (with $\gamma(0)=1$) to $\tilde\gamma:[0,1]\to\mathbb{R}$ (the universal cover) via $e^{2\pi i\tilde\gamma}=\gamma$. The homotopy class of $\gamma$ is determined by $\tilde\gamma(1)\in\mathbb{Z}$ (the winding number). The map $[\gamma]\mapsto\tilde\gamma(1)$ is a group isomorphism $\pi_1(S^1)\xrightarrow{\sim}\mathbb{Z}$: concatenation corresponds to addition of winding numbers.
Example 2
Show $\mathbb{R}^2\setminus\{0\}$ is homotopy equivalent to $S^1$.
Solution: Define $r:\mathbb{R}^2\setminus\{0\}\to S^1$ by $r(x)=x/|x|$ (retraction) and $i:S^1\hookrightarrow\mathbb{R}^2\setminus\{0\}$ (inclusion). $r\circ i=\text{id}_{S^1}$. $i\circ r$ is homotopic to $\text{id}$ via $H(x,t)=(1-t)x+t(x/|x|)$; this is continuous since $|H(x,t)|=(1-t)|x|+t>0$ for $t\in[0,1]$. So $r$ is a homotopy equivalence: $\mathbb{R}^2\setminus\{0\}\simeq S^1$ and $\pi_1(\mathbb{R}^2\setminus\{0\})\cong\mathbb{Z}$.
Practice
- Compute $\pi_1(\mathbb{R}P^2)$ and $\pi_1(T^2)$ using van Kampen's theorem.
- Prove that simply connected spaces have unique path-lifting for covering maps.
- Show $\pi_1(X\times Y)\cong\pi_1(X)\times\pi_1(Y)$ using the product of loops.
- Describe the covering spaces of $S^1$ and relate them to subgroups of $\mathbb{Z}$.
Show Answer Key
1. $\pi_1(\mathbb{R}P^2)\cong\mathbb{Z}/2$ (the double cover $S^2\to\mathbb{R}P^2$ shows the fundamental group has order 2). $\pi_1(T^2)\cong\mathbb{Z}\times\mathbb{Z}$ (the torus is $S^1\times S^1$, and $\pi_1$ of a product is the product of fundamental groups).
2. Let $p:\tilde{X}\to X$ be a covering map, $\tilde{X}$ simply connected. For a path $\gamma$ in $X$ starting at $x_0$, and a lift $\tilde{x}_0\in p^{-1}(x_0)$: the lift $\tilde{\gamma}$ exists (path-lifting lemma) and is unique because any two lifts starting at $\tilde{x}_0$ differ by a loop in $\tilde{X}$, which is null-homotopic ($\tilde{X}$ simply connected), so they're the same.
3. $\pi_1(X\times Y,(x_0,y_0))\cong\pi_1(X,x_0)\times\pi_1(Y,y_0)$. A loop in $X\times Y$ projects to loops in $X$ and $Y$ via $\text{pr}_1,\text{pr}_2$. The map $[\gamma]\mapsto([\text{pr}_1\circ\gamma],[\text{pr}_2\circ\gamma])$ is the desired isomorphism.
4. Covering spaces of $S^1$: $\mathbb{R}\to S^1$ (universal cover, corresponding to trivial subgroup $\{0\}\le\mathbb{Z}$), and $S^1\to S^1$ by $z\mapsto z^n$ (corresponding to $n\mathbb{Z}\le\mathbb{Z}$). These are all the connected covering spaces, one for each subgroup of $\mathbb{Z}=\pi_1(S^1)$.