Fundamental group

Intuition [Beginner]

Pick a point $x_0$ inside a space $X$ and consider every continuous loop that starts at $x_0$, wanders around $X$, and returns to $x_0$. Two loops count as the same if you can continuously deform one into the other while keeping the endpoints pinned at $x_0$. The collection of these "deformation classes" of loops is the fundamental group ${\pi }_1(X,x_0)$, with multiplication given by tracing one loop and then the other.

The fundamental group answers a single, sharp question: which loops in $X$ can be contracted to a point, and how do the non-contractible ones combine? On the plane, every loop shrinks — there are no obstructions. On a circle, a loop that winds twice around cannot be unwound to a loop that winds once: the integer count of how many times you go around is exactly ${\pi }_1(S^1)=\mathbb{Z}$. On a torus, the longitude and meridian loops are independent obstructions: ${\pi }_1({\mathbb{T}}^2)={\mathbb{Z}}^2$.

This unit zooms in on the fundamental group itself — the loop construction, the group operation, the fact that continuous maps move ${\pi }_1$ around in a way that respects composition, and the standard examples computed in detail. The broader picture of homotopy and higher homotopy groups lives in 03.12.01.

Visual [Beginner]

A space with a chosen basepoint $x_0$ and several loops based at $x_0$ — some contractible, some winding around an obstruction. Two loops in the same homotopy class are connected by a continuous family of intermediate loops that all pass through $x_0$ at their endpoints.

The group operation is concatenation: walk the first loop, then the second. The identity is the constant loop that never moves. The inverse is the same loop traversed backwards.

Worked example [Beginner]

Compute the fundamental group of the circle $S^1$ as the integers, by hand. Place the basepoint $x_0=1$ on the unit circle in the complex plane. The loop ${\gamma }_n(s)=e^{2\pi ins}$ for an integer $n$ winds around $S^1$ exactly $n$ times — counterclockwise if $n$ is positive, clockwise if $n$ is negative.

Two loops with the same winding number can be deformed into each other: smoothly slide one onto the other while the winding count stays put. Two loops with different winding numbers cannot: the winding count is a continuous integer-valued function of a continuous family of loops, so it is constant on each homotopy class.

Concatenation adds winding numbers. Walking ${\gamma }_n$ and then ${\gamma }_m$ produces a loop that winds $n+m$ times. The constant loop has winding zero. Reversing ${\gamma }_n$ gives ${\gamma }_{-n}$. So the homotopy classes form a group with the integers as carrier and addition as operation: ${\pi }_1(S^1,1)=\mathbb{Z}$.

What this tells us: the fundamental group is the algebraic shadow of the topological obstructions to contracting loops. The integer $n$ is the winding number, and it is exactly the data that distinguishes loops on a circle.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $(X,x_0)$ be a pointed topological space 02.01.01 02.01.02. A loop at $x_0$ is a continuous map $\gamma :[0,1]\to X$ with $\gamma (0)=\gamma (1)=x_0$. Two loops ${\gamma }_0,{\gamma }_1$ are homotopic rel basepoint if there exists a continuous map $H:[0,1]\times [0,1]\to X$ with

$$H(s,0)={\gamma }_0(s),H(s,1)={\gamma }_1(s),H(0,t)=H(1,t)=x_0.$$

The double endpoint condition $H(0,t)=H(1,t)=x_0$ pins the basepoint throughout the deformation; this is what distinguishes the based version of homotopy from the free version of 03.12.01.

Concatenation. For loops ${\gamma }_1,{\gamma }_2$ at $x_0$, define

$$({\gamma }_1\cdot {\gamma }_2)(s)=\{\begin{array}{ll}{\gamma }_1(2s)& 0\le s\le \frac{1}{2},\\ {\gamma }_2(2s-1)& \frac{1}{2}\le s\le 1.\end{array}$$

The constant loop $c_{x_0}(s)=x_0$ is the candidate identity, and the reversed loop $\bar{\gamma }(s)=\gamma (1-s)$ is the candidate inverse.

Definition. The fundamental group of $(X,x_0)$ is

$${\pi }_1(X,x_0):=\{\text{loops at }{x}_0\}/(\text{homotopy rel basepoint}),$$

with multiplication $[{\gamma }_1]\cdot [{\gamma }_2]:=[{\gamma }_1\cdot {\gamma }_2]$, identity $[c_{x_0}]$, and inverse $[\gamma {]}^{-1}=[\bar{\gamma }]$.

The verification that this is a group requires three steps. Associativity: the loops $({\gamma }_1\cdot {\gamma }_2)\cdot {\gamma }_3$ and ${\gamma }_1\cdot ({\gamma }_2\cdot {\gamma }_3)$ are reparametrisations of one another; an explicit homotopy is built by interpolating linearly between the two reparametrisations of $[0,1]$. Identity: $c_{x_0}\cdot \gamma$ is a reparametrisation of $\gamma$ that pauses at $x_0$ on the first half; the homotopy compresses the pause. Inverse: $\gamma \cdot \bar{\gamma }$ is null-homotopic via the homotopy that retracts the loop to the basepoint along itself, $H(s,t)=\gamma (2s(1-t))$ on $[0,1/2]$ and $H(s,t)=\gamma ((2-2s)(1-t))$ on $[1/2,1]$.

Functoriality. A based continuous map $f:(X,x_0)\to (Y,y_0)$ induces a homomorphism $f_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0)$ by $f_{\ast }[\gamma ]=[f\circ \gamma ]$. Composition is preserved: $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$, and $({\mathrm{id}}_X{)}_{\ast }={\mathrm{id}}_{{\pi }_1(X,x_0)}$.

Homotopy invariance. If $f,g:(X,x_0)\to (Y,y_0)$ are based-homotopic, then $f_{\ast }=g_{\ast }$. Hence ${\pi }_1$ is a homotopy-invariant of pointed spaces and descends to a functor on the homotopy category of pointed spaces.

Basepoint independence (path-connected case). For a path $\eta :x_0\to x_1$ in $X$, define

$${\beta }_{\eta }:{\pi }_1(X,x_0)\to {\pi }_1(X,x_1),{\beta }_{\eta }[\gamma ]=[\bar{\eta }\cdot \gamma \cdot \eta ].$$

Then ${\beta }_{\eta }$ is a group isomorphism with inverse ${\beta }_{\bar{\eta }}$. Two paths $\eta ,{\eta }^{\prime }$ from $x_0$ to $x_1$ produce isomorphisms ${\beta }_{\eta }$, ${\beta }_{{\eta }^{\prime }}$ that differ by an inner automorphism of ${\pi }_1(X,x_1)$. Consequently, for path-connected $X$, the fundamental group is well-defined up to inner automorphism, and one writes ${\pi }_1(X)$ when only the isomorphism class is needed.

A space is simply connected if it is path-connected and ${\pi }_1(X,x_0)=0$ for some (equivalently, every) basepoint.

Key theorem with proof [Intermediate+]

Theorem (${\pi }_1(S^1)=\mathbb{Z}$). The map $\mathrm{deg}:{\pi }_1(S^1,1)\to \mathbb{Z}$ that sends a loop class $[\gamma ]$ to its winding number is a group isomorphism.

The proof uses the universal cover $p:\mathbb{R}\to S^1$, $p(t)=e^{2\pi it}$. The deck-transformation group is $\mathbb{Z}$ acting on $\mathbb{R}$ by integer translation, and $p^{-1}(1)=\mathbb{Z}\subseteq \mathbb{R}$.

Path-lifting lemma. For any path $\gamma :[0,1]\to S^1$ with $\gamma (0)=1$ and any $\tilde{a}\in p^{-1}(1)=\mathbb{Z}$, there is a unique continuous lift $\tilde{\gamma }:[0,1]\to \mathbb{R}$ with $p\circ \tilde{\gamma }=\gamma$ and $\tilde{\gamma }(0)=\tilde{a}$.

Homotopy-lifting lemma. For any homotopy $H:[0,1]\times [0,1]\to S^1$ with $H(\cdot ,0)={\gamma }_0$ and any lift ${\tilde{\gamma }}_0$ of ${\gamma }_0$, there is a unique continuous lift $\tilde{H}:[0,1]\times [0,1]\to \mathbb{R}$ with $p\circ \tilde{H}=H$ and $\tilde{H}(\cdot ,0)={\tilde{\gamma }}_0$.

Both lemmas follow from the local-triviality of $p$ via a Lebesgue-number subdivision of the unit interval (resp. unit square) into pieces small enough that $p$ trivialises over each, and a step-by-step extension of the lift using the unique lift on each piece.

Definition of $\mathrm{deg}$. For a loop $\gamma$ at $1$, take the unique lift $\tilde{\gamma }$ with $\tilde{\gamma }(0)=0$. Set $\mathrm{deg}(\gamma ):=\tilde{\gamma }(1)$, which is an integer because $p(\tilde{\gamma }(1))=\gamma (1)=1$ forces $\tilde{\gamma }(1)\in p^{-1}(1)=\mathbb{Z}$.

Well-definedness. If ${\gamma }_0\simeq {\gamma }_1$ rel basepoint via a homotopy $H$, lift $H$ to $\tilde{H}$ with $\tilde{H}(\cdot ,0)={\tilde{\gamma }}_0$. The endpoint $\tilde{H}(1,t)\in p^{-1}(1)=\mathbb{Z}$ is a continuous integer-valued function of $t$, hence constant. So $\tilde{H}(1,0)=\tilde{H}(1,1)$, which means $\mathrm{deg}({\gamma }_0)=\mathrm{deg}({\gamma }_1)$.

Homomorphism. For loops ${\gamma }_1,{\gamma }_2$ at $1$ with respective lifts ${\tilde{\gamma }}_1,{\tilde{\gamma }}_2$ starting at $0$, the lift of ${\gamma }_1\cdot {\gamma }_2$ starting at $0$ is the concatenation ${\tilde{\gamma }}_1\cdot (\mathrm{deg}({\gamma }_1)+{\tilde{\gamma }}_2)$, where the second piece is translated by $\mathrm{deg}({\gamma }_1)$ so the lift is continuous. Its endpoint is $\mathrm{deg}({\gamma }_1)+{\tilde{\gamma }}_2(1)=\mathrm{deg}({\gamma }_1)+\mathrm{deg}({\gamma }_2)$.

Surjectivity. The loop ${\gamma }_n(s)=e^{2\pi ins}$ lifts to ${\tilde{\gamma }}_n(s)=ns$, with ${\tilde{\gamma }}_n(1)=n$. So $\mathrm{deg}({\gamma }_n)=n$.

Injectivity. If $\mathrm{deg}(\gamma )=0$, the lift $\tilde{\gamma }$ is a loop in $\mathbb{R}$ based at $0$. Since $\mathbb{R}$ is contractible, $\tilde{\gamma }$ is null-homotopic via the straight-line homotopy $\tilde{H}(s,t)=(1-t)\tilde{\gamma }(s)$, which keeps the endpoints fixed at $0$. Composing with $p$ gives a basepoint-fixing homotopy from $\gamma$ to the constant loop. Hence $[\gamma ]=[c_1]$, and $\mathrm{deg}$ is injective.

The map $\mathrm{deg}$ is therefore a group isomorphism ${\pi }_1(S^1,1)\cong \mathbb{Z}$. $\square$

Bridge. The lifting machinery proven here is exactly what builds 03.12.02 (covering space) into the classification theorem: subgroups of ${\pi }_1(X,x_0)$ correspond to connected covering spaces of $X$, with $p:\mathbb{R}\to S^1$ as the prototype example. The same data appears again in 03.12.08 (fundamental groupoid), where ${\pi }_1(X,x_0)$ is recovered as the automorphism group of $x_0$ in the groupoid ${\pi }_1(X)$, and again in 03.12.09 (Seifert-van Kampen) as the central computational tool. Putting these together, the foundational reason this unit precedes the broader homotopy story is exactly that every later result on ${\pi }_n$ for $n\ge 2$ builds on the loop-space identification ${\pi }_n(X)={\pi }_{n-1}(\Omega X)$, which only makes sense once the $n=1$ case has been pinned down independently.

Counterexamples to common slips

• Without basepoint pinning, "${\pi }_1$" collapses. A free homotopy of loops (no constraint on $H(0,t),H(1,t)$) identifies $\gamma$ with any conjugate $\bar{\eta }\cdot \gamma \cdot \eta$. The free-homotopy classes form the conjugacy classes in ${\pi }_1(X,x_0)$, not the fundamental group itself. The basepoint pinning is what gives a group, not just a set with a partial structure.
• Concatenation is not strictly associative. The two parenthesisations $({\gamma }_1\cdot {\gamma }_2)\cdot {\gamma }_3$ and ${\gamma }_1\cdot ({\gamma }_2\cdot {\gamma }_3)$ traverse the three loops at different speeds; they are equal only after a reparametrisation of $[0,1]$. The group law is associative on homotopy classes, not on loops.
• Connectedness is not enough for basepoint independence. The connecting path $\eta :x_0\to x_1$ requires path-connectedness, which is strictly stronger than connectedness. The topologist's sine curve is connected but not path-connected, and there is no canonical isomorphism between fundamental groups at points in different path-components.
• ${\pi }_1({\mathbb{RP}}^n)=\mathbb{Z}/2$ for $n\ge 2$, but ${\pi }_1({\mathbb{RP}}^1)=\mathbb{Z}$. Real projective $1$-space is the circle and inherits ${\pi }_1=\mathbb{Z}$. The $\mathbb{Z}/2$ answer kicks in only once the higher cell collapses the generator into a $2$-torsion element.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none — Mathlib supplies FundamentalGroup as the automorphism group of a basepoint inside FundamentalGroupoid, but the surrounding loop-space API (compact-open topology on $\Omega X$, ${\pi }_1(S^1)=\mathbb{Z}$ via the universal cover, basepoint conjugation, recursive identification with higher homotopy) is not yet in place. The Codex companion module re-exports the existing skeleton and flags the missing pieces.

[object Promise]

The five-piece formalisation roadmap is recorded in lean_mathlib_gap above and feeds the upstream Mathlib contribution queue.

Advanced results [Master]

Loop space. Equip the path space $C([0,1],X)$ with the compact-open topology. The loop space at $x_0$ is the subspace

$$\Omega (X,x_0):=\{\gamma \in C([0,1],X):\gamma (0)=\gamma (1)=x_0\},$$

itself a pointed topological space (with the constant loop as basepoint). Concatenation gives an H-space structure on $\Omega (X,x_0)$, associative and unital up to homotopy. Path-components are exactly homotopy classes:

$${\pi }_0(\Omega (X,x_0))={\pi }_1(X,x_0).$$

The recursive identification ${\pi }_n(X,x_0)={\pi }_{n-1}(\Omega (X,x_0),c_{x_0})$ for $n\ge 1$ wires the fundamental group into the entire higher-homotopy tower of 03.12.01. The free loop space $\mathcal{L}X=C(S^1,X)$ relaxes the basepoint condition; its connected components are conjugacy classes in ${\pi }_1$, and the loop bundle $\Omega (X,x_0)\hookrightarrow \mathcal{L}X\to X$ relates the two.

Galois correspondence (preview, full statement in 03.12.02). Let $X$ be path-connected, locally path-connected, and semi-locally simply connected. Pick a basepoint $x_0$. The map $\tilde{X}\mapsto p_{\ast }({\pi }_1(\tilde{X},{\tilde{x}}_0))\subseteq {\pi }_1(X,x_0)$ from connected pointed covering spaces of $X$ to subgroups of ${\pi }_1(X,x_0)$ is a bijection. Normal subgroups correspond to regular (Galois) covers; the zero subgroup corresponds to the universal cover $\tilde{X}$, on which ${\pi }_1(X,x_0)$ acts by deck transformations.

Specialisation: surface groups. For a closed orientable genus-$g$ surface ${\Sigma }_g$ ($g\ge 1$), the standard $4g$-gon presentation

$${\pi }_1({\Sigma }_g)=⟨a_1,b_1,\dots ,a_g,b_g\mid [a_1,b_1][a_2,b_2]\cdots [a_g,b_g]=1⟩$$

is computed via Seifert-van Kampen 03.12.09 applied to the polygon side identifications, with the single relator coming from the boundary of the $2$-cell. The genus-$0$ case is the sphere, ${\pi }_1=0$; the genus-$1$ case is the torus, ${\pi }_1={\mathbb{Z}}^2$, with the relator $[a,b]=1$ degenerating to the abelianness condition; for $g\ge 2$ the surface group is non-abelian, hyperbolic, and torsion-free.

Functoriality on the homotopy category. The functor ${\pi }_1:{\mathbf{Top}}_{\ast }\to \mathbf{Grp}$ factors through the homotopy category $\mathrm{h}{\mathbf{Top}}_{\ast }$, since homotopic maps induce equal ${\pi }_1$-maps. Restricted to path-connected pointed spaces with abelian fundamental group, ${\pi }_1$ takes values in $\mathbf{Ab}$ and agrees with the first singular homology $H_1$ (Hurewicz, 03.12.19). On non-abelian fundamental groups, the abelianisation ${\pi }_1(X,x_0)\to {\pi }_1(X,x_0{)}^{\mathrm{ab}}$ is the Hurewicz map, and the kernel is the commutator subgroup.

Relation to the fundamental groupoid 03.12.08. The fundamental groupoid ${\pi }_1(X)$ has objects the points of $X$ and morphisms ${\mathrm{Hom}}_{{\pi }_1(X)}(x,y)$ the path-homotopy classes of paths $x\to y$. The fundamental group is exactly the automorphism group at a basepoint:

$${\pi }_1(X,x_0)={\mathrm{Aut}}_{{\pi }_1(X)}(x_0).$$

For path-connected $X$, the inclusion $\{x_0\}\hookrightarrow X$ induces an equivalence of categories $\mathrm{B}{\pi }_1(X,x_0)\simeq {\pi }_1(X)$, where $\mathrm{B}{\pi }_1(X,x_0)$ is the one-object groupoid built from the group. The groupoid framing is more flexible — it survives loss of path-connectedness, where the group framing fails — and is the right setting for Brown's groupoid Seifert-van Kampen theorem 03.12.09.

Synthesis. The fundamental group is the simplest nontrivial homotopy invariant, and every later structure in algebraic topology is in some sense a refinement of it. Read upward, the recursion ${\pi }_n(X)={\pi }_{n-1}(\Omega X)$ identifies the higher homotopy groups as iterated loop spaces of $X$, and the Eckmann-Hilton argument forces them to be abelian for $n\ge 2$. Read downward, the Galois correspondence identifies subgroups of ${\pi }_1$ with covering spaces, the abelianisation is the first homology $H_1$, and the Hurewicz map is the bridge to the singular complex. Read sideways, the groupoid extension ${\pi }_1(X)=$ groupoid of path-homotopy classes recovers ${\pi }_1(X,x_0)$ as ${\mathrm{Aut}}_{{\pi }_1(X)}(x_0)$ and removes the path-connectedness assumption for Seifert-van Kampen. The central insight is that this single object — the group of based loop classes — is exactly the structural signature that the rest of the homotopy strand reads off.

Full proof set [Master]

Group axioms on ${\pi }_1(X,x_0)$. The verification consists of three explicit homotopies on the unit square. Associativity: define $H_{\mathrm{assoc}}:[0,1{]}^2\to X$ as the linear interpolation between the reparametrisations of $[0,1]$ underlying $({\gamma }_1\cdot {\gamma }_2)\cdot {\gamma }_3$ and ${\gamma }_1\cdot ({\gamma }_2\cdot {\gamma }_3)$; explicitly, partition $[0,1]$ into three pieces with breakpoints linear in $t$ between $(\frac{1}{4},\frac{1}{2})$ at $t=0$ and $(\frac{1}{2},\frac{3}{4})$ at $t=1$, and trace ${\gamma }_1,{\gamma }_2,{\gamma }_3$ on the corresponding pieces. Identity: the homotopy from $c_{x_0}\cdot \gamma$ to $\gamma$ shrinks the constant-loop interval $[0,\frac{1}{2}]$ to a point; explicitly, $H(s,t)=c_{x_0}(s^{\prime })$ on $s\le \frac{1-t}{2}$ and $H(s,t)=\gamma \left(\frac{2s-1+t}{1+t}\right)$ on $s\ge \frac{1-t}{2}$. Inverse: $\gamma \cdot \bar{\gamma }$ is null-homotopic via $H(s,t)=\gamma (2s(1-t))$ on $s\le \frac{1}{2}$, $H(s,t)=\gamma ((2-2s)(1-t))$ on $s\ge \frac{1}{2}$, retracting both halves to $x_0$ as $t\to 1$.

Functoriality and homotopy invariance. A based continuous map $f:(X,x_0)\to (Y,y_0)$ takes a loop $\gamma$ at $x_0$ to a loop $f\circ \gamma$ at $y_0$, and the basepoint-fixing condition on a homotopy $H:{\gamma }_0\simeq {\gamma }_1$ rel endpoints transfers verbatim: $f\circ H$ is a basepoint-fixing homotopy from $f\circ {\gamma }_0$ to $f\circ {\gamma }_1$. So $f_{\ast }$ is well-defined on classes. The point-by-point identity $f\circ ({\gamma }_1\cdot {\gamma }_2)=(f\circ {\gamma }_1)\cdot (f\circ {\gamma }_2)$ gives the homomorphism property. Functoriality $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$ is direct from associativity of function composition. For homotopy invariance, given a based homotopy $K:f\simeq g$, the composite $K\circ (\gamma \times \mathrm{id})$ is a based homotopy from $f\circ \gamma$ to $g\circ \gamma$, so $f_{\ast }[\gamma ]=g_{\ast }[\gamma ]$.

Basepoint independence. For $\eta :x_0\to x_1$, the map ${\beta }_{\eta }:[\gamma ]\mapsto [\bar{\eta }\cdot \gamma \cdot \eta ]$ preserves concatenation up to homotopy (the inserted $\eta \cdot \bar{\eta }$ at the join is null-homotopic by the inverse relation just proved). The composite ${\beta }_{\eta }\circ {\beta }_{\bar{\eta }}$ sends $[\gamma ]$ to $[\bar{\eta }\cdot \eta \cdot \gamma \cdot \bar{\eta }\cdot \eta ]=[\gamma ]$, so ${\beta }_{\eta }$ is a group isomorphism with inverse ${\beta }_{\bar{\eta }}$.

${\pi }_1(S^1)=\mathbb{Z}$. Proved in §"Key theorem".

${\pi }_1(S^n)=0$ for $n\ge 2$. Proved in Exercise 6.

${\pi }_1({\Sigma }_g)$. Apply Seifert-van Kampen 03.12.09 to the standard CW decomposition of ${\Sigma }_g$ as a $4g$-gon with sides identified in the pattern $a_1{b}_1{a}_1^{-1}{b}_1^{-1}\cdots a_g{b}_g{a}_g^{-1}{b}_g^{-1}$ and one $2$-cell attached via the boundary word. Cover ${\Sigma }_g$ by an open neighbourhood $U$ of the $1$-skeleton (a wedge of $2g$ circles, ${\pi }_1=F_{2g}$) and an open neighbourhood $V$ of the open $2$-cell (contractible). The intersection $U\cap V$ deformation-retracts to a circle, with ${\pi }_1(U\cap V)=\mathbb{Z}$ generated by the loop around the boundary of the $2$-cell. The amalgamated free product is $F_{2g}\ast \mathbb{Z}/N$, where $N$ is the normal subgroup generated by the boundary word $\prod [a_i,b_i]$, giving the standard surface-group presentation.

${\pi }_0(\Omega X)={\pi }_1(X)$. A path in $\Omega X$ from ${\gamma }_0$ to ${\gamma }_1$ is a continuous map $[0,1]\to \Omega X$, equivalently a continuous map $[0,1]\times [0,1]\to X$ (by the exponential law for the compact-open topology, when $X$ is sufficiently nice — Hausdorff or compactly generated). The basepoint condition on $\Omega X$ forces $H(0,t)=H(1,t)=x_0$, so a path in $\Omega X$ is exactly a basepoint-fixing homotopy in $X$. Hence path-components of $\Omega X$ are homotopy classes of based loops, i.e. ${\pi }_0(\Omega X)={\pi }_1(X)$. The same argument iterated gives ${\pi }_n(X)={\pi }_{n-1}(\Omega X)$, and the H-space structure on $\Omega X$ from concatenation supplies the (abelian, by Eckmann-Hilton, for $n\ge 2$) group operation on ${\pi }_n$.

Connections [Master]

• Topological space 02.01.01 and continuous map 02.01.02. The construction is functorial in the underlying pointed-space data: a continuous map of pointed spaces gives a homomorphism of fundamental groups, and a based homotopy of maps gives the equality of the induced homomorphisms. The fundamental group is therefore an invariant of based homotopy type.

• Homotopy and homotopy group 03.12.01. This unit deepens the ${\pi }_1$ portion of the broader homotopy unit; conversely, 03.12.01 provides the higher homotopy groups ${\pi }_n$ for $n\ge 2$ and the surrounding category-theoretic apparatus. The recursion ${\pi }_n(X,x_0)={\pi }_{n-1}(\Omega X,c_{x_0})$ wires the two together.

• Covering space 03.12.02. Subgroups of ${\pi }_1(X,x_0)$ correspond bijectively to connected pointed covering spaces of $X$ (Galois correspondence). The universal cover corresponds to the zero subgroup; regular covers correspond to normal subgroups; the deck-transformation group of the universal cover is ${\pi }_1(X,x_0)$. The lifting machinery used to compute ${\pi }_1(S^1)=\mathbb{Z}$ in §"Key theorem" is the prototype.

• Fundamental groupoid 03.12.08. The fundamental group is the automorphism group of a basepoint inside the fundamental groupoid: ${\pi }_1(X,x_0)={\mathrm{Aut}}_{{\pi }_1(X)}(x_0)$. For path-connected $X$, the groupoid is equivalent to the one-object groupoid $\mathrm{B}{\pi }_1(X,x_0)$, and the two contain the same homotopy information.

• Seifert-van Kampen 03.12.09. The principal computational tool: for $X=U\cup V$ with $U,V,U\cap V$ open path-connected and $x_0\in U\cap V$, ${\pi }_1(X,x_0)$ is the amalgamated free product ${\pi }_1(U){\ast }_{{\pi }_1(U\cap V)}{\pi }_1(V)$. The figure-eight, surface groups, and graph groups all compute via this rule.

• Hurewicz theorem 03.12.19. The abelianisation ${\pi }_1(X,x_0)\to {\pi }_1(X,x_0{)}^{\mathrm{ab}}$ is naturally isomorphic to the first singular homology $H_1(X;\mathbb{Z})$ for path-connected $X$. The Hurewicz map is the algebraic bridge from the (non-abelian) fundamental group to the (abelian) homology.

Historical & philosophical context [Master]

Henri Poincaré introduced the fundamental group in Analysis Situs (1895, Journal de l'École Polytechnique) ^{[Poincaré 1895]}, using it to distinguish topological spaces that look identical under crude invariants. His motivating problem was the three-dimensional Poincaré conjecture: a closed simply-connected $3$-manifold is the $3$-sphere. Poincaré's framing was already implicitly groupoid (he considered loops at every point and how they conjugate under path-changes), but the explicit formulation as a group attached to a single basepoint, with the conjugation argument for basepoint independence, became standard during the 1920s and 1930s through the work of Reidemeister, Seifert, and Threlfall.

The set-theoretic and categorical reformulation became canonical in the mid-twentieth century, with Eilenberg-Mac Lane (1945) ^{[EilenbergMacLane1945]} and Hurewicz (1935-1936) ^{[Hurewicz1935]} extending the construction to higher dimensions and identifying the abelianisation with first singular homology. Brown (1967) ^[Brown1967] subsequently introduced the fundamental groupoid as the natural generalisation removing the basepoint hypothesis from Seifert-van Kampen.

Bibliography [Master]

[object Promise]