The fundamental group

Theorem

The fundamental group

T-118Home MU-401Threads space · structure

Statement

Let \(X\) be a topological space and let \(x_0 \in X\) be a chosen basepoint. A loop at \(x_0\) is a continuous map \(\gamma : [0,1] \to X\) with \(\gamma(0)=\gamma(1)=x_0\). Two loops \(\gamma_0,\gamma_1\) at \(x_0\) are path-homotopic, written \(\gamma_0 \simeq \gamma_1\), if there is a continuous map \(H:[0,1]\times[0,1]\to X\) with \(H(s,0)=\gamma_0(s)\), \(H(s,1)=\gamma_1(s)\), and \(H(0,t)=H(1,t)=x_0\) for all \(t\in[0,1]\) (the endpoints stay fixed throughout the homotopy). Let \(\pi_1(X,x_0)\) be the set of path-homotopy classes \([\gamma]\) of loops at \(x_0\). Define a binary operation by concatenation, \[ (\gamma_0 \ast \gamma_1)(s) = \begin{cases} \gamma_0(2s), & 0 \le s \le \tfrac12,\\ \gamma_1(2s-1), & \tfrac12 \le s \le 1,\end{cases} \qquad [\gamma_0][\gamma_1] := [\gamma_0 \ast \gamma_1]. \] Then \(\pi_1(X,x_0)\) is a group under this operation, called the fundamental group of \(X\) at \(x_0\); its identity is the class of the constant loop \(e_{x_0}\), and \([\gamma]^{-1} = [\bar\gamma]\) where \(\bar\gamma(s) = \gamma(1-s)\). Moreover, a continuous based map \(f:(X,x_0)\to(Y,y_0)\) induces a group homomorphism \(f_\ast : \pi_1(X,x_0)\to\pi_1(Y,y_0)\), \(f_\ast[\gamma] = [f\circ\gamma]\), satisfying \((\mathrm{id}_X)_\ast = \mathrm{id}\) and \((g\circ f)_\ast = g_\ast \circ f_\ast\); consequently a homeomorphism (indeed any based homotopy equivalence) of pointed spaces induces an isomorphism of fundamental groups.

Why it matters

The fundamental group is the first, and most computable, member of the family of homotopy invariants that convert topological questions into algebraic ones. It detects "holes" a loop cannot be shrunk across: \(\pi_1(\mathbb{R}^2\setminus\{0\}) \cong \mathbb{Z}\) records that a loop winding once around the puncture cannot be contracted, while \(\pi_1(\mathbb{R}^2)\) is trivial. Because it is a genuine group (not just a set or a pointed set, as \(\pi_0\) is), it carries structural information — abelianness, torsion, presentations — that can be compared using the full machinery of group theory.

Functoriality is what makes the invariant usable: to show two spaces are not homeomorphic (or not even homotopy equivalent) it suffices to show their fundamental groups are not isomorphic, without ever having to rule out all possible continuous maps between the spaces directly. This is the template every later invariant in algebraic topology (higher homotopy groups, homology, cohomology) follows.

Hypotheses

The interval \([0,1]\) carries the standard Euclidean topology and loops are continuous maps \([0,1]\to X\). Counterexample sketch: if "loop" were allowed to mean an arbitrary (not necessarily continuous) function with \(\gamma(0)=\gamma(1)=x_0\), concatenation classes would carry no topological information at all — e.g. every discontinuous bijection \([0,1]\to X\) with matching endpoints would count as a "loop," collapsing the theory and destroying homotopy invariance, since continuity is exactly what homeomorphisms/homotopy equivalences preserve.

Homotopies fix the basepoint throughout (\(H(0,t)=H(1,t)=x_0\) for all \(t\)), not just at \(t=0,1\). Counterexample sketch: on \(X=S^1\), if endpoints were allowed to move freely during the homotopy (free homotopy of loops, not based homotopy) then every loop is freely homotopic to the constant loop by sliding it around, collapsing the free-homotopy-class set to a point and destroying the group structure — free homotopy classes of loops correspond instead to conjugacy classes in \(\pi_1\), not to \(\pi_1\) itself.

The basepoint \(x_0\) is fixed and, without further assumption, \(\pi_1(X,x_0)\) is a group attached to the pair \((X,x_0)\), not to \(X\) alone. Counterexample sketch (subtle): if \(X\) is not path-connected, basepoints in different path-components give unrelated, generally non-isomorphic groups — e.g. \(X = S^1 \sqcup \{\text{pt}\}\) has \(\pi_1(X,x_0)\cong\mathbb{Z}\) for \(x_0\) on the circle but \(\pi_1(X,x_0)\) trivial for \(x_0\) the isolated point. Only under the extra hypothesis that \(X\) is path-connected does a change of basepoint yield an isomorphism (conjugation by a path), so that one may speak loosely of "the fundamental group of \(X\)."

Proof

\(\ast\) is well defined on homotopy classes: if \(\gamma_0\simeq\gamma_0'\) and \(\gamma_1\simeq\gamma_1'\) then \(\gamma_0\ast\gamma_1 \simeq \gamma_0'\ast\gamma_1'\).

Let \(H,K:[0,1]^2\to X\) be based homotopies \(\gamma_0\simeq\gamma_0'\) and \(\gamma_1\simeq\gamma_1'\). Define \(G(s,t)=H(2s,t)\) for \(s\le\tfrac12\) and \(G(s,t)=K(2s-1,t)\) for \(s\ge\tfrac12\). At \(s=\tfrac12\) both formulas give \(H(1,t)=x_0=K(0,t)\), so \(G\) is well defined and continuous by the pasting lemma (gluing two continuous maps agreeing on a closed common boundary). \(G(\cdot,0)=\gamma_0\ast\gamma_1\), \(G(\cdot,1)=\gamma_0'\ast\gamma_1'\), and \(G(0,t)=H(0,t)=x_0\), \(G(1,t)=K(1,t)=x_0\), so \(G\) is a based homotopy. A

Path-homotopy \(\simeq\) is an equivalence relation on loops at \(x_0\).

Reflexivity: \(H(s,t)=\gamma(s)\) (constant in \(t\)) is continuous and based. Symmetry: if \(H\) witnesses \(\gamma_0\simeq\gamma_1\) then \((s,t)\mapsto H(s,1-t)\) witnesses \(\gamma_1\simeq\gamma_0\), continuous as a composite of \(H\) with the continuous map \((s,t)\mapsto(s,1-t)\). Transitivity: if \(H:\gamma_0\simeq\gamma_1\) and \(K:\gamma_1\simeq\gamma_2\), glue in \(t\) exactly as in Step 1 (now the gluing variable is \(t\) instead of \(s\)): \(L(s,t)=H(s,2t)\) for \(t\le\tfrac12\), \(L(s,t)=K(s,2t-1)\) for \(t\ge\tfrac12\); this agrees on \(t=\tfrac12\) since \(H(s,1)=\gamma_1(s)=K(s,0)\), so \(L\) is continuous by the pasting lemma and is a based homotopy \(\gamma_0\simeq\gamma_2\). Hence \(\pi_1(X,x_0)\) as a set of equivalence classes is legitimately defined before Step 1 is even applied. C

\(\ast\) is associative on \(\pi_1(X,x_0)\): \(([\gamma_0][\gamma_1])[\gamma_2] = [\gamma_0]([\gamma_1][\gamma_2])\).

The loops \((\gamma_0\ast\gamma_1)\ast\gamma_2\) and \(\gamma_0\ast(\gamma_1\ast\gamma_2)\) traverse \(\gamma_0,\gamma_1,\gamma_2\) in the same order but at different speeds, governed by a piecewise-linear increasing reparametrization \(\varphi:[0,1]\to[0,1]\) with \(\varphi(0)=0,\varphi(1)=1\) (explicitly, \(\varphi\) sends \([0,\tfrac14],[\tfrac14,\tfrac12],[\tfrac12,1]\) linearly onto \([0,\tfrac12],[\tfrac12,\tfrac34],[\tfrac34,1]\)) such that \((\gamma_0\ast\gamma_1)\ast\gamma_2 = \gamma_0\ast(\gamma_1\ast\gamma_2)\circ\varphi\). The straight-line homotopy \(H(s,t) = \big(\gamma_0\ast(\gamma_1\ast\gamma_2)\big)\big((1-t)\varphi(s) + t s\big)\) is continuous (composite of continuous maps, jointly continuous in \((s,t)\) since the reparametrizing formula is a polynomial in \(s,t\)), fixes \(H(0,t)=x_0\), \(H(1,t)=x_0\), and interpolates between the two concatenations at \(t=0,1\). This is the general reparametrization lemma: any two loops differing by a continuous increasing self-homeomorphism of \([0,1]\) fixing endpoints are path-homotopic via the straight-line homotopy between the reparametrizing maps. B

The class \(e := [\mathrm{const}_{x_0}]\) is a two-sided identity: \([\gamma]\ast e = [\gamma] = e \ast [\gamma]\).

\(\gamma\ast \mathrm{const}_{x_0}\) traverses \(\gamma\) on \([0,\tfrac12]\) at double speed then sits at \(x_0\); it equals \(\gamma\circ\varphi\) for the reparametrization \(\varphi(s)=\min(2s,1)\), which is continuous, increasing, and fixes \(0,1\). By the reparametrization lemma of Step 3, \(\gamma\circ\varphi \simeq \gamma\) via the straight-line homotopy \(H(s,t)=\gamma\big((1-t)\varphi(s)+ts\big)\). Symmetrically for \(\mathrm{const}_{x_0}\ast\gamma\) with \(\varphi(s)=\max(2s-1,0)\). A

Every class has an inverse: \([\gamma]\ast[\bar\gamma] = e = [\bar\gamma]\ast[\gamma]\), where \(\bar\gamma(s)=\gamma(1-s)\).

Define \(H(s,t) = \gamma\big(\rho(s,t)\big)\) where \(\rho(s,t) = \begin{cases}2s(1-t), & 0\le s\le \tfrac12,\\ (2-2s)(1-t), & \tfrac12\le s\le 1.\end{cases}\) This \(\rho\) is continuous (the two formulas agree at \(s=\tfrac12\), giving \((1-t)\)) and jointly continuous in \((s,t)\), so \(H\) is continuous as a composite. At \(t=0\), \(\rho(s,0)\) rises \(0\to1\) then falls \(1\to0\), giving exactly \(\gamma\ast\bar\gamma\); at \(t=1\), \(\rho(s,1)\equiv 0\), giving the constant loop at \(\gamma(0)=x_0\); and \(\rho(0,t)=\rho(1,t)=0\) for all \(t\), so \(H(0,t)=H(1,t)=x_0\) throughout — the homotopy is based. Hence \([\gamma\ast\bar\gamma] = e\). The identity \([\bar\gamma\ast\gamma]=e\) follows by the same construction applied to \(\bar\gamma\) (noting \(\overline{\bar\gamma}=\gamma\)), or directly by running \(\rho\) backwards. B

\((\pi_1(X,x_0), \ast)\) is a group.

Step 1 shows \(\ast\) is a well-defined total binary operation on the set \(\pi_1(X,x_0)\) of equivalence classes (legitimate by Step 2); Step 3 gives associativity; Step 4 gives a two-sided identity \(e\); Step 5 gives two-sided inverses for every element. These are precisely the group axioms. A

Functoriality: for continuous based \(f:(X,x_0)\to(Y,y_0)\), the map \(f_\ast[\gamma]:=[f\circ\gamma]\) is a well-defined group homomorphism \(\pi_1(X,x_0)\to\pi_1(Y,y_0)\), and \((g\circ f)_\ast = g_\ast\circ f_\ast\), \((\mathrm{id}_X)_\ast=\mathrm{id}\).

Well-defined: if \(H\) is a based homotopy \(\gamma_0\simeq\gamma_1\) in \(X\) then \(f\circ H\) is continuous (composite of continuous maps) and is a based homotopy \(f\circ\gamma_0 \simeq f\circ\gamma_1\) in \(Y\), since \(f\circ H(0,t)=f(x_0)=y_0\) and likewise at \(s=1\). Homomorphism: \(f\circ(\gamma_0\ast\gamma_1) = (f\circ\gamma_0)\ast(f\circ\gamma_1)\) holds on the nose (same piecewise formula, just post-composed with \(f\)), so \(f_\ast([\gamma_0][\gamma_1]) = f_\ast[\gamma_0]\,f_\ast[\gamma_1]\). Functor identities: \((g\circ f)\circ\gamma = g\circ(f\circ\gamma)\) as functions, and \(\mathrm{id}_X\circ\gamma=\gamma\), both trivially. This exhibits \(\pi_1\) as a functor from pointed topological spaces and based continuous maps to groups. C

Result

\(\big(\pi_1(X,x_0),\ast\big)\) is a group, and \(\pi_1(-,-)\) is a functor from pointed spaces to groups.

Reading. Loops at a fixed basepoint, glued end to end and considered only up to continuous deformation that keeps the basepoint fixed, obey exactly the group axioms: gluing is associative up to deformation, the loop that never moves is an identity up to deformation, and running a loop backwards undoes it up to deformation. Continuous based maps between spaces translate faithfully into homomorphisms between these groups.

Scope. Applies to any topological space \(X\) with any chosen basepoint \(x_0\) — no local niceness (manifold, CW, Hausdorff) is required for the group structure itself, only continuity of loops and homotopies on \([0,1]\) and \([0,1]^2\). Path-connectedness of \(X\) is not needed to define \(\pi_1(X,x_0)\), only to relate \(\pi_1\) at different basepoints (see Corollaries).

Corollaries & converses

If \(X\) is path-connected, any path \(\delta\) from \(x_0\) to \(x_1\) induces an isomorphism \(\beta_\delta:\pi_1(X,x_1)\to\pi_1(X,x_0)\), \([\gamma]\mapsto[\delta\ast\gamma\ast\bar\delta]\); hence "the fundamental group of a path-connected space" is well defined up to (non-canonical, in general) isomorphism.
A based homotopy equivalence \(f:(X,x_0)\to(Y,y_0)\) (with based homotopy inverse \(g\)) induces an isomorphism \(f_\ast\) by functoriality: \(g_\ast f_\ast = (\mathrm{id}_X)_\ast=\mathrm{id}\) and \(f_\ast g_\ast=\mathrm{id}\). In particular homeomorphic pointed spaces have isomorphic fundamental groups — this is the standard tool for distinguishing spaces (e.g. \(\pi_1(S^1)\cong\mathbb{Z}\) shows \(S^1\) is not simply connected, hence not contractible).
Converse fails: isomorphic fundamental groups do not imply homotopy equivalent spaces. \(S^2\) and a point both have trivial \(\pi_1\), yet \(S^2\) is not contractible (detected instead by \(H_2\) or \(\pi_2\)); the fundamental group is one invariant among many, not a complete one.
If \(X\) is simply connected in the strong sense \(\pi_1(X,x_0)=\{e\}\) for path-connected \(X\), then every loop is null-homotopic and, more generally, any two paths between the same two points are path-homotopic (a fact underlying, e.g., the well-definedness of the winding number and of covering-space monodromy only in the trivial case).

Fails without

Drop continuity of the homotopy (allow arbitrary set-theoretic \(H\)): every loop becomes "homotopic" to every other loop with the same endpoints via a discontinuous \(H\), collapsing \(\pi_1(X,x_0)\) to a single point for any \(X\) with at least one loop — the invariant carries zero information and is useless for distinguishing spaces such as \(S^1\) (winding number \(1\)) from a point.
Drop the fixed-basepoint condition on the homotopy (use free homotopy of loops instead): the set of free homotopy classes on \(X=S^1\) is a single point (every loop slides around to the constant loop), whereas \(\pi_1(S^1,x_0)\cong\mathbb{Z}\) — the based theorem's group structure is lost entirely, since free homotopy classes correspond only to conjugacy classes in \(\pi_1\) and in general carry no group operation (concatenation of freely homotopic loops is not well defined, because the basepoint can differ after sliding).
Drop path-connectedness when asserting basepoint-independence: for \(X=\{a\}\sqcup S^1\) (disjoint union of a point and a circle), \(\pi_1(X,a)\) is trivial while \(\pi_1(X,x_0)\cong\mathbb{Z}\) for \(x_0\) on the circle — there is no isomorphism relating them because no path in \(X\) connects \(a\) to \(x_0\), so the phrase "the fundamental group of \(X\)" (without reference to a basepoint) is meaningless here.

Common errors

Writing \(\gamma_0\ast\gamma_1 = \gamma_1\ast\gamma_0\) as loops (not merely as homotopy classes, and not even that in general): concatenation is not commutative on the nose, and \(\pi_1\) itself need not be abelian (e.g. \(\pi_1\) of a figure-eight, or of the complement of two points in \(\mathbb{R}^2\), is free of rank 2, nonabelian).
Confusing based homotopy of loops with free homotopy of loops, and concluding \(\pi_1(S^1)\) is trivial by "sliding the loop off"; free homotopy is a coarser relation used for conjugacy classes, not for the group \(\pi_1\).
Treating \((\gamma_0\ast\gamma_1)\ast\gamma_2\) and \(\gamma_0\ast(\gamma_1\ast\gamma_2)\) as literally equal functions \([0,1]\to X\); they are only path-homotopic (different parametrizing speeds), which is precisely why associativity needs Step 3's reparametrization argument rather than being immediate.
Forgetting the basepoint entirely and asserting "\(X\) has fundamental group \(G\)" for a non-path-connected \(X\), or applying the change-of-basepoint isomorphism of the Corollaries without checking path-connectedness.
Assuming \(f_\ast\) is injective or surjective for an arbitrary continuous \(f\); functoriality only gives a homomorphism, and only a homotopy equivalence guarantees an isomorphism.

Discussion

The fundamental group was introduced by Poincaré (1895, Analysis Situs) as one of the first algebraic invariants attached to a space up to continuous deformation, predating the systematic homology theory that would eventually subsume and extend it. Its construction is a paradigm case of a recurring move in topology: take a geometric object (loops), quotient by the "right" equivalence (homotopy rel basepoint), and discover that a natural geometric operation (concatenation) descends to an algebraic one (group multiplication) on the quotient.

The proof above is entirely a proof about reparametrizations of the interval: every group axiom (associativity, identity, inverses) is verified by exhibiting an explicit, jointly continuous "straight-line" homotopy between two ways of traversing \([0,1]\) at different speeds, then transporting it through \(\gamma\). This is why the argument works verbatim for any space \(X\) whatsoever — none of it uses local structure of \(X\), only that loops and homotopies are continuous maps out of \([0,1]\) and \([0,1]^2\), and that composition of continuous maps is continuous (used repeatedly, together with the pasting/gluing lemma for functions agreeing on a shared closed boundary).

Functoriality (Step 7) is the deeper structural payoff: it upgrades \(\pi_1\) from "a group associated to each pointed space" to a genuine functor \(\mathbf{Top}_\ast \to \mathbf{Grp}\), which is what licenses the standard proof technique "compute \(\pi_1\) of both spaces and compare" for showing spaces are topologically or homotopically distinct — including proofs of the Brouwer fixed-point theorem in dimension 2, the fundamental theorem of algebra, and the Borsuk–Ulam theorem, all of which route through \(\pi_1(S^1)\cong\mathbb{Z}\) and functoriality rather than direct geometric argument.

Common misconception: that \(\pi_1(X,x_0)\) somehow "is" the set of loops, or that homotopy classes can be visualized as literally finitely many loops; in general \(\pi_1(X,x_0)\) can be infinite, even uncountable-index in its conjugacy structure for wild spaces, and elements are equivalence classes under a relation defined by the existence of a homotopy, not by any canonical representative.

Worked examples

Show directly from the group axioms (not by computing \(\pi_1(S^1)\) itself) that if \([\gamma]\ast[\gamma] = [\gamma]\) in \(\pi_1(X,x_0)\) for some loop \(\gamma\), then \(\gamma\) is null-homotopic, i.e. \([\gamma]=e\).

Purely a group-theory computation using Step 6: multiply both sides of \([\gamma]\ast[\gamma]=[\gamma]\) on the left by the inverse \([\gamma]^{-1}=[\bar\gamma]\) guaranteed by Step 5. A

\([\gamma]^{-1}\ast\big([\gamma]\ast[\gamma]\big) = [\gamma]^{-1}\ast[\gamma]\)

Both sides equal because we started from the hypothesis \([\gamma]\ast[\gamma]=[\gamma]\) and composed with the same class \([\gamma]^{-1}\) on the left; well-definedness of \(\ast\) on classes (Step 1) makes this a legitimate substitution. A

\(\big([\gamma]^{-1}\ast[\gamma]\big)\ast[\gamma] = e\), i.e. \(e\ast[\gamma]=e\), i.e. \([\gamma]=e\)

Left side regrouped by associativity (Step 3), then \([\gamma]^{-1}\ast[\gamma]=e\) by Step 5, then \(e\) is a two-sided identity by Step 4. This uses only the group axioms established for \(\pi_1(X,x_0)\), independent of what \(X\) actually is. A

In any group, \(g^2=g \implies g=e\); applied to \(\pi_1(X,x_0)\), an "idempotent" loop class is trivial.

Reading. The fundamental group theorem is what makes this a meaningful statement about loops at all: without Steps 1–6 establishing genuine group axioms, "\([\gamma]\ast[\gamma]=[\gamma]\)" would be a statement about a mere set with an operation, and cancellation would not be licensed.

Let \(X = \mathbb{R}^2\), any basepoint \(x_0\). Show \(\pi_1(\mathbb{R}^2,x_0) = \{e\}\), using the theorem's group structure together with an explicit contraction.

\(\mathbb{R}^2\) is convex, so for any loop \(\gamma\) at \(x_0\) the straight-line homotopy \(H(s,t) = (1-t)\gamma(s) + t\,x_0\) is a continuous map \([0,1]^2\to\mathbb{R}^2\) (jointly continuous, being an affine combination of the continuous \(\gamma\) and constants). B

\(H(s,0)=\gamma(s)\), \(H(s,1)=x_0\), \(H(0,t)=(1-t)x_0+tx_0=x_0\), \(H(1,t)=(1-t)x_0+tx_0=x_0\).

Direct evaluation of the affine formula at \(t=0,1\) and at \(s=0,1\) (using \(\gamma(0)=\gamma(1)=x_0\)); the based-homotopy condition of the Statement is exactly \(H(0,t)=H(1,t)=x_0\), which holds for every \(t\). A

Hence \(\gamma \simeq \mathrm{const}_{x_0}\), so \([\gamma]=e\) for every loop \(\gamma\); since \(\gamma\) was arbitrary, \(\pi_1(\mathbb{R}^2,x_0)\) has only the identity element.

By definition of \(\pi_1\) as the set of path-homotopy classes (Statement) and the fact that \(e\) is the class of the constant loop (Step 4 of the Proof); the theorem guarantees \(\pi_1(\mathbb{R}^2,x_0)\) is a group at all, and this computation shows it is the trivial group. A

\(\pi_1(\mathbb{R}^2,x_0) = \{e\}\); more generally \(\pi_1\) of any convex (or star-shaped) subset of \(\mathbb{R}^n\) is trivial.

Reading. Convexity gives an explicit straight-line contraction of every loop to the basepoint; the fundamental group theorem is what guarantees this "count" of loops-up-to-deformation is a well-defined group rather than an ad hoc set, so "trivial group" is an unambiguous, algebraically meaningful conclusion (as opposed to, say, on \(S^1\), where the same style of question yields \(\mathbb{Z}\), not a point).

Problems

Prove that if \(x_0,x_1\) lie in the same path component of \(X\) and \(\delta\) is a path from \(x_0\) to \(x_1\), then \(\beta_\delta([\gamma]) := [\delta\ast\gamma\ast\bar\delta]\) defines a group isomorphism \(\pi_1(X,x_1)\to\pi_1(X,x_0)\).
Solution

Well-defined: if \(\gamma\simeq\gamma'\) rel endpoints in \(X\) via \(H\), then \((s,t)\mapsto \delta\ast H(\cdot,t)\ast\bar\delta\) evaluated appropriately gives a based homotopy \(\delta\ast\gamma\ast\bar\delta \simeq \delta\ast\gamma'\ast\bar\delta\) at \(x_0\) (concatenate the constant-in-\(t\) paths \(\delta,\bar\delta\) with the homotopy \(H\) in the middle slot, using the well-definedness of \(\ast\) on homotopy classes, Step 1 of the Proof, applied fibrewise in \(t\)). Homomorphism: \(\beta_\delta([\gamma_0])\ast\beta_\delta([\gamma_1]) = [\delta\ast\gamma_0\ast\bar\delta]\ast[\delta\ast\gamma_1\ast\bar\delta] = [\delta\ast\gamma_0\ast\bar\delta\ast\delta\ast\gamma_1\ast\bar\delta]\). Since \(\bar\delta\ast\delta \simeq \mathrm{const}_{x_1}\) (Step 5 of the Proof, applied to the path \(\delta\)), and concatenating with a class-\(e\) loop leaves the class unchanged (Step 4), this collapses to \([\delta\ast\gamma_0\ast\gamma_1\ast\bar\delta] = \beta_\delta([\gamma_0]\ast[\gamma_1])\). Inverse: \(\beta_{\bar\delta}\) is a two-sided inverse to \(\beta_\delta\) by the same cancellation argument applied to \(\delta\ast\bar\delta\simeq\mathrm{const}_{x_0}\) and \(\bar\delta\ast\delta\simeq\mathrm{const}_{x_1}\). Hence \(\beta_\delta\) is a bijective homomorphism, i.e. an isomorphism.
Let \(f,g:(X,x_0)\to(Y,y_0)\) be continuous based maps that are based-homotopic (\(f\simeq g\) rel \(x_0\)). Show \(f_\ast = g_\ast : \pi_1(X,x_0)\to\pi_1(Y,y_0)\).
Solution

Let \(K:X\times[0,1]\to Y\) be the based homotopy, \(K(\cdot,0)=f\), \(K(\cdot,1)=g\), \(K(x_0,t)=y_0\) for all \(t\). For a loop \(\gamma\) at \(x_0\), define \(J(s,t) = K(\gamma(s),t)\), a composite of continuous maps \([0,1]\times[0,1]\xrightarrow{\gamma\times\mathrm{id}} X\times[0,1]\xrightarrow{K} Y\), hence continuous. Then \(J(s,0)=K(\gamma(s),0)=f(\gamma(s)) = (f\circ\gamma)(s)\), \(J(s,1)=(g\circ\gamma)(s)\), and \(J(0,t)=K(x_0,t)=y_0=J(1,t)\). So \(J\) is a based homotopy \(f\circ\gamma \simeq g\circ\gamma\), giving \(f_\ast[\gamma] = [f\circ\gamma] = [g\circ\gamma] = g_\ast[\gamma]\) for every \([\gamma]\), i.e. \(f_\ast=g_\ast\) as functions on \(\pi_1(X,x_0)\).
Give an example of a space \(X\) and basepoint \(x_0\) for which \(\pi_1(X,x_0)\) is nonabelian, and identify (without full proof) which hypothesis of commutativity is simply absent from the group axioms verified in the Proof.
Solution

Take \(X\) to be a wedge of two circles (figure eight), \(x_0\) the wedge point. By the Seifert–van Kampen theorem (a further result, not proved here), \(\pi_1(X,x_0)\) is the free group on two generators \(F_2 = \langle a,b\rangle\), which is nonabelian: \(ab\ne ba\) as words, since no relation forces them equal. The Proof of T-118 (Steps 1–6) never used or needed commutativity of \(\ast\): associativity, identity, and inverses are the only axioms verified, and indeed the reparametrization argument for associativity (Step 3) works regardless of whether \(\gamma_0\ast\gamma_1\) and \(\gamma_1\ast\gamma_0\) are homotopic — in general they are not.
Using only the group axioms established in the Proof (not the value of \(\pi_1(S^1)\)), show that in \(\pi_1(X,x_0)\), the identity element is unique, i.e. if \(e'\) also satisfies \([\gamma]\ast e' = [\gamma] = e'\ast[\gamma]\) for all \([\gamma]\), then \(e'=e\).
Solution

Apply the hypothesis on \(e'\) to \([\gamma]=e\): \(e\ast e' = e\). But \(e\) is a two-sided identity (Step 4 of the Proof), so \(e\ast e' = e'\). Combining, \(e' = e\ast e' = e\). This is the standard group-theoretic uniqueness-of-identity argument, applicable here precisely because Step 6 established that \(\pi_1(X,x_0)\) satisfies the group axioms.
Let \(X\) be path-connected and suppose \(\pi_1(X,x_0) = \{e\}\) for some (hence, by Problem 1, every) \(x_0\). Show that any two paths \(\alpha,\beta:[0,1]\to X\) with the same endpoints \(\alpha(0)=\beta(0)=p\), \(\alpha(1)=\beta(1)=q\) are path-homotopic rel endpoints.
Solution

Consider the loop \(\gamma = \alpha \ast \bar\beta\) based at \(p\) (well defined since \(\alpha(1)=q=\beta(1)\), so \(\bar\beta(0)=q=\alpha(1)\), matching the concatenation formula). Since \(\pi_1(X,p)=\{e\}\), \([\gamma]=e\), i.e. \(\alpha\ast\bar\beta \simeq \mathrm{const}_p\) rel endpoints. Concatenate both sides on the right with \(\beta\) and use the reparametrization/associativity and identity facts from Steps 3–5 of the Proof: \((\alpha\ast\bar\beta)\ast\beta \simeq \alpha\ast(\bar\beta\ast\beta) \simeq \alpha\ast \mathrm{const}_q \simeq \alpha\), while \(\mathrm{const}_p\ast\beta \simeq \beta\). Since path-homotopy is transitive (Step 2) and homotopies compose under concatenation (Step 1's well-definedness), \(\alpha \simeq \beta\) rel endpoints.