Fundamental group (${\pi }_1(X,x_0)$)

The fundamental group ${\pi }_1(X,x_0)$ consists of based loops modulo based homotopy. Multiplication follows one based loop and then the other; the constant based loop represents the identity, and path reversal gives the inverse. Passing to homotopy classes makes this operation associative, as explained by well-defined loop concatenation.

Identify the circle with $S^1=\{z\in \mathbb{C}:|z|=1\}$ and choose basepoint $1$. Its based loopshave winding number $n$: positive $n$ counts counterclockwise turns, negative $n$ counts clockwise turns, and $n=0$ gives the constant based loop. Every based loop on the circle is joined by a based homotopy to exactly one ${\gamma }_n$. ConsequentlyThis is Hatcher's Theorem 1.7 in Algebraic Topology, §1.1, pp. 29–30. For example, two counterclockwise turns followed by one clockwise turn represent $2+(-1)=1$, rather than three turns. The winding graph in the figure above makes the signed arithmetic visible.

A proof outline uses the covering map $q:\mathbb{R}\to S^1$, $q(u)=e^{2\pi iu}$. By the path lifting theorem, a based loop has a unique lift starting at zero and ending at an integer $n$. Interpolating its lift to $s\mapsto ns$ produces a based homotopy to ${\gamma }_n$. The homotopy lifting theorem for a covering map keeps that endpoint unchanged during a based homotopy, proving uniqueness.

In the filled disk, every based loop $\gamma$ at $x_0$ has a based homotopy to the constant based loop:Both endpoints remain $x_0$, and the convex set property of the disk keeps every intermediate point inside. Thus ${\pi }_1(D^2,x_0)$ is the trivial group. Together with the deformation retraction above, this computes three concrete answers: the filled disk has trivial fundamental group, while the circle and the annulus have infinite cyclic groups.

Maps have computable algebraic shadows too. The continuous map $p_n:S^1\to S^1$, $p_n(z)=z^n$, fixes $1$ and sends ${\gamma }_k$ to ${\gamma }_{nk}$. Its induced group homomorphism is therefore $k\mapsto nk$, and its mapping degree is $n$. For $n=2$ the image consists of even integers, so this map cannot be a homotopy equivalence. Composing $p_2$ with $p_{-3}$ gives $p_{-6}$: map composition multiplies degrees, whereas concatenating based loops adds winding numbers. Distinct degrees distinguish these maps up to homotopy; this calculation does not classify arbitrary topological spaces.

Historically, Poincaré's 1895 Analysis Situs introduced the fundamental group; its original §12 explicitly names it. The CNRS transcription of Poincaré's §12, Groupe fondamental provides a direct view of this early formulation.

Two based edge loops are equivalent under insertion or deletion of stationary steps, cancellation or insertion of a backtracking spur, and replacement of two sides of a triangle by its third side. The edge-loop group consists of the resulting classes, with concatenation as product and reversal as inverse. Each elementary move can be performed inside a longer concatenation, so multiplication is well defined. Associativity follows from concatenation, the stationary loop is the identity, and a loop followed by its reversal cancels by spurs.

Realization induces a homomorphism to the fundamental group of the geometric realization, because each move is a homotopy in an edge or a triangle. The relative Simplicial approximation theorem applied to based circles makes this map surjective. The full edge-path theorem additionally proves injectivity by approximating homotopies; that stronger assertion is not needed for the group construction or the surjectivity argument.

For a finite connected graph, collapse a spanning tree. Its $v-1$ edges disappear, and each of the remaining $e-v+1$ edges gives a circle in a bouquet of circles. The fundamental group of a bouquet of circles is free group, with one member of a free basis of a group for each circle. This gives the formula in terms of the Euler characteristic $\chi (X)=v-e$.

The fundamental group of a surface. For a closed orientable genus-$g$ surface the fundamental group of a closed orientable surface has $2g$ generators and one relation, the product of their paired commutators.

Removing an embedded open disk from the torus leaves a topological surface with one boundary circle. In the square model, radial expansion away from the removed disk gives a deformation retraction onto the square boundary with opposite sides identified, a wedge of two circles. Its fundamental group is therefore the free group on two group generators.

In the standard free basis of a group $a,b$ for the fundamental group of a once-punctured torus, its oriented boundary loop represents $ab{a}^{-1}{b}^{-1}$ up to orientation and conjugation. The square model proves this: radial deformation retraction takes the inner boundary once around the four sides, whose orientations read $a,b,a^{-1},b^{-1}$. Attaching a simply connected space along that boundary kills the normal closure of the commutator of group elements, whenever the hypotheses of the Seifert-van Kampen theorem hold.

For a closed orientable aspherical topological surface, every outer automorphism of its fundamental group is induced by a surface homeomorphism. The mapping class group allowing orientation reversal is therefore the full outer automorphism group. For the torus the statement is realized by linear maps in $G{L}_2(\mathbb{Z})$; for higher genus it is the Dehn-Nielsen-Baer theorem. A semidirect product by the infinite cyclic group is consequently the fundamental group of the corresponding mapping torus.

Every finitely presented group is the fundamental group of a closed connected smooth manifold in dimension at least four. Build a compact zero/one/two-handle manifold $W$ with the generators and relators of the presentation. Disjoint attaching loops can be embedded in its boundary of dimension at least three. The Seifert-van Kampen theorem gives ${\pi }_1(W)=T$, and the dual relative handle structure makes ${\pi }_1(\partial W)\to T$ surjective. Doubling $W$ along its boundary gives the amalgam of two copies of $T$ along that same surjection, which is again $T$. This construction underlies binary families of nonhomeomorphic Sunada quotients.

A simply connected space is path connected and has trivial fundamental group: every closed loop contracts to a point. The three-sphere is simply connected, whereas Real projective space ${\mathbb{RP}}^3$ has fundamental group ${\mathbb{Z}}_2$.

The fundamental group of a bouquet of $r$ circles is the free group on $r$ generators, represented by the oriented circles. For finite $r$, iterated Seifert-van Kampen theorem proves the claim by attaching one circle at a time. Nontrivial reduced words in a free group distinguish based loops, while free homotopy classes correspond to conjugacy classes.

Let $u$ be a path from $x_0$ to $x_1$. Traversing paths from left to right gives an isomorphismConcatenation and cancellation of a path with its reverse prove the homomorphism property and show that $\bar{u}$ supplies its inverse. A different connecting path changes this isomorphism by an inner automorphism, so it need not be canonical.

Concatenating two based homotopies proves that loop concatenation depends only on based-homotopy classes. Endpoint-fixing reparametrizations provide associativity and the identity laws on classes.

A continuous based 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 composition of loops with $f$.

For a suitably path-connected open cover $X=U\cup V$, the fundamental group of $X$ is the pushout of the homomorphisms from ${\pi }_1(U\cap V)$ to ${\pi }_1(U)$ and ${\pi }_1(V)$.

Attach a closed interval to a sphere $S^2$ at two distinct endpoints, with its interior disjoint from the sphere. The fundamental group is infinite cyclic group. Remove a point of the sphere away from the endpoints: the remaining plane has a deformation retraction onto an arc between the endpoints, keeping the added closed interval fixed. This gives a circle. An open cap about the removed point is simply connected, and its punctured overlap has trivial map into the first fundamental group because it lies in the plane. The Seifert-van Kampen theorem therefore adds no further relator.

Forthe Seifert-van Kampen theorem gives

Given homomorphisms $i_A:C\to A$ and $i_B:C\to B$, the amalgamated free product $A{\ast }_{C}B$ comes with homomorphisms from $A$ and $B$ that agree on $C$. It is universal with this property: every compatible pair of homomorphisms $A\to H$ and $B\to H$ factors uniquely through $A{\ast }_{C}B$.

The free product $A\ast B$ is the amalgamated free product over the trivial group. Its nonidentity elements have unique reduced alternating normal forms.

Let $G=⟨x,y\mid y^2=1⟩$. For each $n\ge 2$, a free basis of a subgroup of rank $n$ isA covering-space proof uses $r=n-1$ spheres double-covering the real projective plane, with antipodal points $a_j,b_j$ over the wedge point. Add lifted $x$ edges in a cyclic order through the $a_j$, and an $x$ loop at each $b_j$. This is a connected $2r$-sheeted cover of a circle wedged with a projective plane. After collapsing one connecting arc in each simply connected sphere, its fundamental group is that of a graph with $r$ vertices and $2r$ edges, hence free of rank $r+1$.

A spanning tree along the first $r-1$ cyclic edges gives the displayed words: the closing cycle is $x^r$, and the loop reached through $a_j,b_j$ is $x^{j}yxy{x}^{-j}$. Injectivity of the induced fundamental-group map of a covering map proves they form a free basis in $G$.

Suppose $P$ has a soluble word problem for a group and $z$ generates an infinite cyclic group. In $P\ast ⟨z⟩$, for $g\ne 1$ the words $gz$ and $[g,z]$ are cyclically reduced with two and four syllables respectively. Their nonzero powers have syllable lengths $2|k|$ and $4|k|$. Given an input's reduced free product form, its length bounds the possible exponents; compare the finitely many candidates using the base word algorithm. Membership and the exponent in each of these infinite cyclic subgroups are therefore computable, with no algorithm for the order of $g$ required.

A subgroup $A$ is a free factor of $G$ if $G$ is isomorphic to a free product $A\ast B$ by an isomorphism restricting to the inclusion of $A$, for some subgroup $B$. The normal form theorem for a free product ensures that both factors embed. A free factor need not be a normal subgroup.

Given group homomorphisms $f_A:A\to H$ and $f_B:B\to H$, there is exactly one group homomorphism $A\ast B\to H$ extending both. Its value on an alternating normal form is the product of the factor-map values. Thus the free product is the coproduct in the category of groups.

A free product $A\ast B$ is nontrivial as a splitting when both factors differ from the trivial group. Alternating words $(ab{)}^r$ with $a,b\ne 1$ in different factors show that every such splitting gives an infinite group. The distinct normal forms $ab$ and $ba$ also show that the group is not an abelian group.

For nontrivial groups $A,B$, the Cayley graph of $A\ast B$ with $S=(A\setminus \{1\})\cup (B\setminus \{1\})$ has the edges of the Bass-Serre tree as its vertices. Two such vertices are adjacent exactly when the tree edges share an endpoint. Mapping each group element to its tree-edge midpoint is an isometry on the vertex metrics and has image within distance $1/2$ of every tree point, hence gives a quasi-isometry of metric graphs. The normal form theorem for a free product identifies $|g{|}_S$ with reduced syllable length. The factor subgroups have diameter one in this metric.

A normal form for free groups and free product of groups represents elements by reduced words. In a free group, adjacent inverse letters cancel; in a free product, alternating nonidentity syllables from different factors are unique. In an amalgamated free product, chosen coset representatives produce the corresponding reduced form relative to the common subgroup.

Every element of a free product $G\ast H$ has a unique expression $u_1\cdots u_n$ in which each $u_i$ is a nonidentity element of one factor and adjacent syllables belong to different factors. The identity corresponds to the empty word.

A syllable is a nonidentity element of one factor in a free product word. A reduced normal form has successive syllables in different factors; its syllable length is the number of these elements. This length records factor elements, rather than letters in a chosen generating set.

If $X=U_1\cup U_2$, the sets $U_1,U_2$, and their intersection are path-connected, and the base point lies in the intersection, then ${\pi }_1(X)$ is generated by the images of ${\pi }_1(U_1)$ and ${\pi }_1(U_2)$. Subdivide each loop into arcs lying in cover members and join every transition point to the base point within the intersection; the inserted paths telescope.

A Möbius band retracts to its core circle, and its boundary wraps twice around the core. Attaching it to $X$ along a loop representing $w\in {\pi }_1(X)$ therefore adjoins a generator $x$ with relation $x^2=w$.

Glue the boundary of a Möbius band to the projective line in the real projective plane by a mapping degree $n$. The band is the mapping cylinder of the degree-two boundary covering of its core. Collapsing the connecting tree in the resulting double mapping cylinder gives a CW complex with one vertex, edges $a,b$, and two attaching words $a^2$ and $a^{-n}{b}^2$. Its cellular chain complex has degree-two boundary matrix $\left(\begin{array}{cc}2& -n\\ 0& 2\end{array}\right)$. Its determinant is four, so $H_2=0$. The Smith normal form gives $H_1\cong \mathbb{Z}/4$ for odd $n$ and $(\mathbb{Z}/2{)}^2$ for even $n$; $H_0=\mathbb{Z}$ and higher homology vanishes. The space cannot be homotopy equivalent to a closed manifold: mod-two homology forces a potential manifold to be a surface, and its Euler characteristic is one, whereas the only such closed connected surface is the projective plane, with different integral $H_1$.

Attach Möbius bands to a torus along the classes $ab$ and $a^2{b}^3$. If their core generators are $x,y$, the resulting presentation isBecause the exponent vectors $(1,1)$ and $(2,3)$ form a unimodular basis of ${\mathbb{Z}}^2$, eliminating $a,b$ givesThis group maps onto $S_3$ by $x\mapsto (12)$ and $y\mapsto (23)$, so it is nonabelian.

Attaching a disc to $X$ along a based loop $\alpha$ kills precisely the normal closure of its homotopy class:

A group presentation gives a two-dimensional cell complex with one vertex, one oriented loop for each generator, and one 2-cell attached along each relator. Its fundamental group is the presented group.

Attach three 2-cells to $S^1\vee S^1$ along loops representing $a^2$, $b^3$, and $(ab{)}^2$. The resulting group isIndeed, $a\mapsto (12)$ and $b\mapsto (123)$ gives a surjection to $S_3$, while $aba=b^{-1}$ reduces every word to one of $b^i$ or $a{b}^i$, so the presented group has at most six elements.

For the closed orientable surface of genus $g$,In particular, the sphere has trivial fundamental group and the torus has fundamental group ${\mathbb{Z}}^2$.