The fundamental group · Qual Corpus

For a space $X$ and $x_0\in X$, $\pi_1(X,x_0)$ is the set of homotopy classes, relative to the endpoints, of loops at $x_0$, with the group operation induced by concatenation. A path $\gamma$ from $x_0$ to $x_1$ induces an isomorphism $\pi_1(X,x_0)\to\pi_1(X,x_1)$, $[\alpha]\mapsto[\overline\gamma\cdot\alpha\cdot\gamma]$, which depends on $\gamma$ only through its homotopy class; different paths give isomorphisms that differ by conjugation. A path-connected space is simply connected if its fundamental group is trivial, and convex subsets of $\RR^n$ are simply connected through straight-line homotopies. The Seifert–van Kampen theorem: if $X=U\cup V$ with $U$ and $V$ open, and $U$, $V$, and $U\cap V$ path-connected and containing $x_0$, then $\pi_1(X,x_0)\cong\pi_1(U,x_0)*_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0)$. The fundamental group of the wedge of $n$ circles is the free group on $n$ generators.

Path (Tag D-J6XOC)

Let $X$ be a topological space and $I\coloneqq[0,1]$. A path in $X$ is a continuous map $\gamma\colon I\to X$; it is a path from $\gamma(0)$ to $\gamma(1)$. A path $\gamma$ is a loop if $\gamma(0) = \gamma(1)$, and then it is a loop based at $\gamma(0)$. For paths $\gamma, \eta\colon I\to X$ with $\gamma(1) = \eta(0)$, the concatenation $\gamma\cdot\eta\colon I\to X$ is the path \[ (\gamma\cdot\eta)(s)\coloneqq \begin{cases} \gamma(2s) & 0\leq s\leq 1/2, \\ \eta(2s-1) & 1/2\leq s\leq 1. \end{cases} \]

See [@Hat02] and [@Mun00].

Fundamental group (Tag D-EBNUE)

Let $X$ be a topological space and $x_0\in X$, and let $I = [0,1]$. The fundamental group $\pi_1(X, x_0)$ is the set of classes of loops $\gamma\colon I\to X$ based at $x_0$ under homotopy rel endpoints, with multiplication \[ [\gamma][\eta]\coloneqq[\gamma\cdot\eta], \] where $\gamma\cdot\eta$ is the concatenation of $\gamma$ and $\eta$.

The multiplication on $\pi_1(X, x_0)$ is well defined and makes $\pi_1(X, x_0)$ a group. Its identity is the class of the constant loop at $x_0$, and $\inverseof{[\gamma]} = [\bar\gamma]$, where $\bar\gamma(s)\coloneqq\gamma(1-s)$.

Let $h\colon I\to X$ be a path from $x_0$ to $x_1$, with reverse $\bar h$. Then $\beta_h\colon\pi_1(X, x_1)\to\pi_1(X, x_0)$, $\beta_h[\gamma]\coloneqq[h\cdot\gamma\cdot\bar h]$, is a group isomorphism. It depends only on the homotopy class of $h$ rel endpoints, and for a second path $h'$ from $x_0$ to $x_1$, $\beta_{h'} = c\circ\beta_h$, where $c$ is conjugation by $[h'\cdot\bar h]\in\pi_1(X, x_0)$. In particular, if $X$ is path connected, the groups $\pi_1(X, x)$ for $x\in X$ are all isomorphic.

See [@Hat02].

Simply connected space (Tag D-GFM35)

A topological space $X$ is simply connected if $X$ is path connected and the fundamental group $\pi_1(X, x_0)$ is trivial for every $x_0\in X$.

For a path connected space $X$, the following are equivalent:

$X$ is simply connected;
every continuous map $\gamma\colon S^1\to X$ is homotopic to a constant map;
every continuous map $\gamma\colon S^1\to X$ extends to a continuous map $\hat\gamma\colon D^2\to X$, that is, $\hat\gamma|_{S^1} = \gamma$;
any two paths $p_1, p_2\colon[0,1]\to X$ with $p_1(0) = p_2(0)$ and $p_1(1) = p_2(1)$ are homotopic rel endpoints.

See [@Hat02].

$\pi_1$ detects simple connectivity (Tag PR-EWJMJ)

A path-connected space is simply connected exactly when its fundamental group is trivial.Suggest a slogan

Let $X$ be a path-connected space. Then $X$ is simply connected if and only if $\pi_1(X, x_0) = 1$ for some, equivalently every, basepoint $x_0\in X$ [@Hat02].

Maps into a convex set are homotopic (Tag T-2YTCZ)

Convex targets make any two maps homotopic by straight-line interpolation.Suggest a slogan

Let $X$ be a topological space and $C\subseteq\RR^n$ a convex subset. Any two continuous maps $f, g\colon X\to C$ are homotopic, via the linear homotopy $F(x,t) = (1-t)f(x) + t\,g(x)$, which lies in $C$ by convexity. For $X = [0,1]$ and paths with common endpoints, the linear homotopy is a homotopy rel endpoints [@Hat02].

Free product of groups (Tag D-JDDCP)

Let $(G_\alpha)_{\alpha\in A}$ be a family of groups. A reduced word is a finite sequence $g_1g_2\cdots g_k$, $k\geq 0$, in which each letter $g_i$ lies in $G_{\alpha_i}\sm\ts{1}$ for some $\alpha_i\in A$ and $\alpha_i\neq\alpha_{i+1}$ for $1\leq i<k$. The free product $\ast_{\alpha\in A} G_\alpha$ is the set of reduced words, with product given by concatenating two words and then reducing: adjacent letters from the same $G_\alpha$ are replaced by their product in $G_\alpha$, and letters equal to $1$ are deleted, until the word is reduced.

The free product $\ast_\alpha G_\alpha$ is a group whose identity is the empty word, and each $G_\beta$ is a subgroup via one-letter words. For every group $H$ and homomorphisms $\varphi_\alpha\colon G_\alpha\to H$ there is a unique homomorphism $\varphi\colon\ast_\alpha G_\alpha\to H$ restricting to $\varphi_\alpha$ on each $G_\alpha$, so $\ast_\alpha G_\alpha$ is the coproduct of the $G_\alpha$ in $\Grp$.

For a set $S$, the free group on $S$ is isomorphic to $\ast_{s\in S}\ZZ$.

See [@Hat02].

Free product with amalgamation (Tag D-WSFYS)

Let $A$, $B$, and $C$ be groups, and let $i\colon C\to A$ and $j\colon C\to B$ be homomorphisms. The free product with amalgamation of $A$ and $B$ over $C$ is \[ A \ast_C B \coloneqq \qty{A \ast B} / N, \] where $A\ast B$ is the free product and $N$ is the normal subgroup of $A\ast B$ generated by $\ts{ i(c) \inverseof{j(c)} \st c\in C }$.

Together with the homomorphisms $A\to A\ast_C B$ and $B\to A\ast_C B$ induced by the inclusions of $A$ and $B$ into $A\ast B$, the group $A\ast_C B$ is the pushout of $A\xleftarrow{i} C\xrightarrow{j} B$ in $\Grp$.

By van Kampen's theorem, if $X = U_1\union U_2$ with $U_1$, $U_2$ and $U_1\intersect U_2$ open and path-connected, then $\pi_1(X)$ is the free product of $\pi_1(U_1)$ and $\pi_1(U_2)$ amalgamated over $\pi_1(U_1\intersect U_2)$ [@Hat02].

[@Hat02].

Seifert--van Kampen theorem (Tag T-BTPU4)

The fundamental group of a union is the pushout of the fundamental groups over the intersection.Suggest a slogan

Let $X = U_{1} \union U_{2}$, where $U_1$, $U_2$ and $U_1\intersect U_2$ are open, path-connected and nonempty, and let $x_0 \in U_1\intersect U_2$. Let $\iota_k\colon \pi_1(U_1\intersect U_2, x_0)\to\pi_1(U_k, x_0)$, $k = 1, 2$, be induced by the inclusions. Then the inclusions $U_1, U_2\injects X$ induce an isomorphism \[ \pi_{1}(U_1, x_0) \ast_{\pi_{1}(U_1 \intersect U_2, x_0)} \pi_{1}(U_2, x_0) \xrightarrow{\ \sim\ } \pi_{1}(X, x_0) \] onto $\pi_1(X, x_0)$ from the free product with amalgamation, which is the pushout of $\pi_1(U_1, x_0)\xleftarrow{\iota_1}\pi_1(U_1\intersect U_2, x_0)\xrightarrow{\iota_2}\pi_1(U_2, x_0)$ in the category of groups [@Hat02].

Let $X = U_1\union U_2$, where $U_1$, $U_2$ and $U_1\intersect U_2$ are open, path-connected and nonempty, let $x_0\in U_1\intersect U_2$, and let $\iota_k\colon \pi_1(U_1\intersect U_2, x_0)\to\pi_1(U_k, x_0)$, $k = 1, 2$, be induced by the inclusions. Suppose \[ \pi_{1}(U_1, x_0) = \left\langle u_{1}, \ldots, u_{k} \suchthat \alpha_{1}, \ldots, \alpha_{l}\right\rangle, \qquad \pi_{1}(U_2, x_0) = \left\langle v_{1}, \ldots, v_{m} \suchthat \beta_{1}, \ldots, \beta_{n}\right\rangle, \] and that $w_1, \ldots, w_p$ generate $\pi_1(U_1\intersect U_2, x_0)$. Then \[ \pi_{1}(X, x_0) \cong \left\langle u_{1}, \ldots, u_{k}, v_{1}, \ldots, v_{m} \suchthat \alpha_{1}, \ldots, \alpha_{l},\ \beta_{1}, \ldots, \beta_{n},\ \iota_1(w_{1}) \iota_2(w_{1})^{-1}, \ldots, \iota_1(w_{p}) \iota_2(w_{p})^{-1} \right\rangle \] [@Hat02].

The hypothesis that $U_1 \intersect U_2$ is path-connected is necessary: for $S^1$ covered by two open arcs, the intersection has two components, both arcs are simply connected, and $\pi_1(S^1)\cong\ZZ$.

$\pi_1$ of a wedge sum (Tag PR-3APOT)

If each basepoint is a deformation retract of an open neighborhood, $\pi_1$ of a wedge of path-connected spaces is the free product of their fundamental groups.Suggest a slogan

Let $(X, x_0)$ and $(Y, y_0)$ be path-connected pointed spaces such that $x_0$ and $y_0$ are deformation retracts of open neighborhoods in $X$ and $Y$ respectively. Then the inclusions induce an isomorphism \[ \pi_1(X)\ast\pi_1(Y)\xrightarrow{\ \sim\ }\pi_1(X \vee Y) \] [@Hat02].

$\pi_1$ of a product (Tag PR-OO3DH)

Fundamental groups turn products of path-connected spaces into direct products.Suggest a slogan

Let $X$ and $Y$ be path-connected spaces. Then the projections induce an isomorphism \[ \pi_1 (X \cross Y) \xrightarrow{\ \sim\ } \pi_1(X) \cross \pi_1(Y) \] [@Hat02].

Homotopy groups (Tag D-EUX36)

Let $X$ be a topological space, $x_0\in X$, and $n\geq 1$, and let $I^n = [0,1]^n$ with boundary $\del I^n$. The $n$th homotopy group $\pi_n(X, x_0)$ is the set of homotopy classes of maps $f\colon(I^n, \del I^n)\to(X, x_0)$, where the homotopies $f_t$ are required to satisfy $f_t(\del I^n) = \ts{x_0}$ for all $t$, with the operation $[f]+[g]\coloneqq[f+g]$ given by \[ (f+g)(s_1, s_2, \ldots, s_n)\coloneqq \begin{cases} f(2s_1, s_2, \ldots, s_n) & 0\leq s_1\leq 1/2, \\ g(2s_1-1, s_2, \ldots, s_n) & 1/2\leq s_1\leq 1. \end{cases} \]

For $n\geq 1$ the operation on $\pi_n(X, x_0)$ is well defined and makes $\pi_n(X, x_0)$ a group, and $\pi_1(X, x_0)$ is the fundamental group. For $n\geq 2$ the group $\pi_n(X, x_0)$ is abelian. Since $I^n/\del I^n\cong S^n$, the set $\pi_n(X, x_0)$ is in bijection with the set $[(S^n, s_0), (X, x_0)]$ of based homotopy classes of based maps $(S^n, s_0)\to(X, x_0)$.

See [@Hat02].

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