The Seifert–van Kampen theorem

Theorem

The Seifert–van Kampen theorem

T-119Home MU-401Threads space · structure

Statement

Let \(X\) be a topological space and let \(U, V \subseteq X\) be open subsets with \(X = U \cup V\), such that \(U\), \(V\) and \(U \cap V\) are all path-connected and nonempty. Fix a basepoint \(x_0 \in U \cap V\). Let \(i_U : U \cap V \to U\), \(i_V : U \cap V \to V\), \(j_U : U \to X\), \(j_V : V \to X\) be the inclusion maps. Then the homomorphism \[ \Phi : \pi_1(U,x_0) \ast \pi_1(V,x_0) \longrightarrow \pi_1(X,x_0) \] induced by \((j_U)_\ast\) and \((j_V)_\ast\) on the free product is surjective, and its kernel is the normal subgroup \(N\) generated by all elements \(i_U{}_\ast(\gamma)\, i_V{}_\ast(\gamma)^{-1}\) for \(\gamma \in \pi_1(U \cap V, x_0)\). Equivalently, \[ \pi_1(X,x_0) \;\cong\; \pi_1(U,x_0) \,\ast_{\pi_1(U\cap V,x_0)}\, \pi_1(V,x_0), \] the pushout of \(\pi_1(U,x_0) \xleftarrow{(i_U)_\ast} \pi_1(U\cap V,x_0) \xrightarrow{(i_V)_\ast} \pi_1(V,x_0)\) in the category of groups, i.e. the group with presentation obtained by taking presentations of \(\pi_1(U,x_0)\) and \(\pi_1(V,x_0)\) and adjoining the relations \(i_U{}_\ast(\gamma) = i_V{}_\ast(\gamma)\) for a generating set of \(\gamma \in \pi_1(U \cap V, x_0)\).

Why it matters

The fundamental group is defined via loops, a construction with no obvious way to "glue" — a loop in a union is not generally a loop in either piece. The Seifert–van Kampen theorem is the mechanism that repairs this: it converts an open cover of a space into an algebraic gluing recipe (a pushout of groups), turning geometric decomposition into computable group presentations. It is the single most productive tool for computing fundamental groups by hand: wedges of circles, surfaces from polygon identifications, complements of points and curves, and CW complexes all yield to it.

It is also the prototype for a much larger pattern in mathematics — the idea that a global invariant of a space glued from pieces is a colimit of the invariants of the pieces (compare Mayer–Vietoris in homology, or descent in sheaf theory). Understanding van Kampen well is understanding the shape of that pattern in its simplest, most concrete form.

Hypotheses

\(U\) and \(V\) are open in \(X\), and \(X = U \cup V\). Openness is what lets the Lebesgue number lemma subdivide an arbitrary path or homotopy into finitely many pieces each landing entirely in \(U\) or entirely in \(V\); the proof's covering argument breaks down for arbitrary (e.g. closed) decompositions. Example: writing \(S^1\) as the union of two closed semicircles meeting only at two points gives no compactness-based subdivision argument, and indeed applying the "formula" naively there gives a wrong answer. \(U\), \(V\) are path-connected. If \(U\) (say) is disconnected, \(\pi_1(U,x_0)\) only sees the path-component of \(x_0\) and the theorem's pushout does not capture the rest of \(X\)'s topology; the map \(\Phi\) can fail to be surjective. E.g. take \(X = \mathbb{R}\), \(U = (-2,-1)\cup(1,2)\)... (excluded because we also need \(U\cap V\) path-connected below, but the point is that dropping path-connectedness of the pieces breaks the surjectivity argument, which explicitly parametrises loops by paths staying inside \(U\) or \(V\)). \(U \cap V\) is path-connected. This is the hypothesis that is easiest to drop carelessly. Classic failure: \(X = S^1\), \(U\), \(V\) the two open arcs each covering slightly more than a semicircle, so \(U \cap V\) is two disjoint short arcs (not path-connected). Both \(U\) and \(V\) are contractible, so the naive pushout formula would predict \(\pi_1(S^1) \cong 1 \ast_{\pi_1(U\cap V)} 1 = 1\), which is false: \(\pi_1(S^1) \cong \mathbb{Z}\). The correct statement when \(U \cap V\) is disconnected requires the fundamental groupoid on a set of basepoints meeting every component of \(U \cap V\) (Problem 3 below works this case in full). \(x_0 \in U \cap V\) is a single common basepoint. All four fundamental groups in the statement must be based at the same point for the maps \(i_U{}_\ast, i_V{}_\ast, (j_U)_\ast, (j_V)_\ast\) to be defined without choosing paths; since \(U\cap V\) is path-connected this involves no loss of generality (change-of-basepoint isomorphisms handle any other choice), but omitting it makes the statement of the pushout ill-posed.

Proof

\(\Phi\) is a well-defined group homomorphism.

A loop or homotopy-class of loops in \(U\) (resp. \(V\)) based at \(x_0\) pushes forward to \(X\) via \((j_U)_\ast\) (resp. \((j_V)_\ast\)), each itself well defined since post-composition with a continuous map sends homotopic loops to homotopic loops. The free product \(\pi_1(U,x_0)\ast\pi_1(V,x_0)\) is the coproduct in the category of groups, so by its universal property the two homomorphisms \((j_U)_\ast,(j_V)_\ast\) into \(\pi_1(X,x_0)\) glue uniquely to a homomorphism \(\Phi\) out of the free product, sending a word \(\alpha_1\beta_1\alpha_2\beta_2\cdots\) (\(\alpha_i \in \pi_1(U,x_0)\), \(\beta_i\in\pi_1(V,x_0)\)) to \((j_U)_\ast(\alpha_1)(j_V)_\ast(\beta_1)\cdots \in \pi_1(X,x_0)\). A

For every \(\gamma \in \pi_1(U\cap V,x_0)\), \(\ \Phi\big(i_U{}_\ast(\gamma)\,i_V{}_\ast(\gamma)^{-1}\big) = 1\); hence \(N \subseteq \ker\Phi\), so \(\Phi\) factors through a homomorphism \(\bar\Phi : \big(\pi_1(U,x_0)\ast\pi_1(V,x_0)\big)/N \to \pi_1(X,x_0)\).

By functoriality of \(\pi_1\), \((j_U)_\ast \circ i_U{}_\ast = (j_V)_\ast \circ i_V{}_\ast\) as both equal \((j_U i_U)_\ast = (j_V i_V)_\ast\), the map induced by the common inclusion \(U\cap V \hookrightarrow X\). So \(\Phi(i_U{}_\ast(\gamma)) = \Phi(i_V{}_\ast(\gamma))\) for every \(\gamma\), i.e. \(\Phi\) kills every generator of \(N\); since \(\ker\Phi\) is normal and contains these generators it contains all of \(N\), and the first isomorphism theorem lets \(\Phi\) descend to the quotient. A

Lebesgue number lemma (cited). If \(K\) is a compact metric space and \(\{A_i\}\) an open cover of \(K\), there is \(\delta \gt 0\) such that every subset of \(K\) of diameter \(\lt \delta\) lies entirely in some \(A_i\).

Standard point-set topology lemma, used twice below (on the interval \([0,1]\) and on the square \([0,1]^2\), both compact metric spaces), to reduce a global statement about a continuous map to a finite combinatorial statement about a fine grid. B

\(\Phi\) is surjective (equivalently \(\bar\Phi\) is surjective): every loop \(f:[0,1]\to X\) based at \(x_0\) is homotopic, rel endpoints, to a finite product of loops each entirely contained in \(U\) or entirely contained in \(V\).

The preimages \(f^{-1}(U), f^{-1}(V)\) form an open cover of the compact metric space \([0,1]\); by Step 3 choose a partition \(0=t_0\lt t_1\lt\cdots\lt t_n=1\) fine enough that each \(f([t_{k-1},t_k])\) lies wholly in \(U\) or wholly in \(V\), and (merging adjacent subintervals of the same type) that consecutive subintervals alternate between the two. For each interior point \(t_k\), \(f(t_k)\) lies in \(U\cap V\) whenever it is a junction between an interval in \(U\) and one in \(V\) (it lies in both closures, and — after a further Lebesgue refinement so that each \(f(t_k)\) itself has a neighbourhood on both sides landing in \(U\cap V\) — actually in \(U\cap V\)); since \(U\cap V\) is path-connected, choose for each such \(k\) a path \(\eta_k\) in \(U\cap V\) from \(x_0\) to \(f(t_k)\) (and \(\eta_0=\eta_n=\) constant path at \(x_0\)). Then \(f\) is homotopic rel endpoints to the concatenation \(\prod_k \big(\eta_{k-1}\cdot f|_{[t_{k-1},t_k]}\cdot \eta_k^{-1}\big)\), a product of loops each lying entirely in \(U\) or entirely in \(V\) (standard reparametrisation/subdivision argument for path concatenation). Hence \([f] = \Phi(w)\) for the corresponding word \(w\) in \(\pi_1(U,x_0)\ast\pi_1(V,x_0)\). B

Setup for injectivity: suppose \(w \in \pi_1(U,x_0)\ast\pi_1(V,x_0)\) is a word \(\alpha_1\beta_1\cdots\alpha_m\beta_m\) with \(\Phi(w)=1\), i.e. the concatenated loop \(f = a_1 b_1 \cdots a_m b_m\) (representatives of \(\alpha_i,\beta_i\)) is null-homotopic in \(X\). We must show \(w \in N\), i.e. \(w\) becomes trivial already in the quotient by the relations \(i_U{}_\ast(\gamma)=i_V{}_\ast(\gamma)\).

This is the substantive half of the theorem: surjectivity only used subdivision of a single loop, but injectivity requires tracking a whole null-homotopy \(H:[0,1]^2\to X\) with \(H(s,0)=f(s)\), \(H(s,1)=H(0,t)=H(1,t)=x_0\), and showing that its combinatorics forces \(w\) into \(N\). C

Grid subdivision of the homotopy: applying Step 3 to the open cover \(\{H^{-1}(U),H^{-1}(V)\}\) of the compact square \([0,1]^2\), choose partitions \(0=s_0\lt\cdots\lt s_p=1\) and \(0=t_0\lt\cdots\lt t_q=1\) fine enough that \(H\) sends each closed rectangle \(R_{ij}=[s_{i-1},s_i]\times[t_{j-1},t_j]\) entirely into \(U\) or entirely into \(V\); refine further so this grid also restricts on \(t=0\) to a partition refining the one used in Step 4 to build \(w\).

Same Lebesgue number lemma argument as Step 4, applied on \([0,1]^2\) with its product metric in place of \([0,1]\); "refine" is legitimate since any common refinement of two finite partitions is again finite and still satisfies the diameter bound. C

For each grid vertex \((s_i,t_j)\) that borders both a \(U\)-rectangle and a \(V\)-rectangle, \(H(s_i,t_j) \in U\cap V\); choose (once and for all, respecting the choices already made on the boundary \(t=0\)) a path in \(U\cap V\) from \(x_0\) to \(H(s_i,t_j)\) for every grid vertex, constant at \(x_0\) on vertices already forced there by the boundary conditions.

Path-connectedness of \(U\cap V\) again (finitely many vertices, so finitely many choices, each independently justified as in Step 4); vertices on the boundary of the square (\(t=0\), \(t=1\), \(s=0\), \(s=1\)) inherit the constant path at \(x_0\) where \(H\) is already \(x_0\) there, and on \(t=0\) inherit the paths already chosen in Step 4. C

Reading off the grid gives a finite sequence of words \(w = w_0, w_1,\dots, w_r = 1\) in \(\pi_1(U,x_0)\ast\pi_1(V,x_0)\), where \(w_{k+1}\) is obtained from \(w_k\) by one of: (a) freely reducing/inserting a trivial factor; (b) replacing a letter by a path-homotopic loop entirely inside the same piece (\(U\) or \(V\)); (c) replacing a factor \(i_U{}_\ast(\gamma)\) by \(i_V{}_\ast(\gamma)\) for the same \(\gamma\in\pi_1(U\cap V,x_0)\), corresponding to crossing one grid rectangle of \(H\) whose loop boundary lies in \(U\cap V\).

Each rectangle \(R_{ij}\) of the grid provides, via its four bounding edges and the corner-to-basepoint paths of Step 7, a homotopy (rel endpoints, inside \(U\) or inside \(V\) according to which piece \(R_{ij}\) maps into) between the loop read along the bottom-and-side edges of \(R_{ij}\) and the loop read along the top-and-side edges; sweeping row by row from \(t=0\) (where the boundary word is \(w\), by construction matching Step 4) to \(t=1\) (where \(H(\cdot,1)\equiv x_0\), giving the trivial word) produces the chain \(w_0,\dots,w_r\). Moves of type (a)/(b) do not change the class of \(w_k\) modulo \(N\) since they are exactly the relations already present in \(\pi_1(U,x_0)\ast\pi_1(V,x_0)\) or already imposed inside a single factor; a move of type (c) occurs precisely when a rectangle straddles the \(U\)/\(V\) boundary, forcing the corresponding sub-loop into \(U\cap V\), and changes \(w_k\) by exactly a generator of \(N\). This is the technical heart of the theorem (full combinatorial bookkeeping: Hatcher, Algebraic Topology, proof of Thm 1.20). C

Hence \(w \equiv w_0 \equiv w_1 \equiv \cdots \equiv w_r = 1 \pmod{N}\), i.e. \(w \in N\); so \(\ker\Phi \subseteq N\), and combined with Step 2, \(\ker\Phi = N\).

Each step \(w_k \to w_{k+1}\) in Step 8 changes the image in the quotient \(\big(\pi_1(U,x_0)\ast\pi_1(V,x_0)\big)/N\) not at all, so \(w \bmod N = w_r \bmod N = 1\). This is exactly injectivity of \(\bar\Phi\) from Step 2. C

Conclusion: \(\bar\Phi : \big(\pi_1(U,x_0)\ast\pi_1(V,x_0)\big)/N \xrightarrow{\ \cong\ } \pi_1(X,x_0)\), and the quotient on the left is by definition the pushout \(\pi_1(U,x_0)\ast_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0)\).

Steps 4 and 9 give surjectivity and injectivity of \(\bar\Phi\) respectively, so it is an isomorphism; the identification of the quotient with the categorical pushout of groups is the standard universal-property description of an amalgamated free product (it satisfies the same mapping-out property that a pushout must by construction, since any pair of homomorphisms out of \(\pi_1(U,x_0)\), \(\pi_1(V,x_0)\) agreeing on \(\pi_1(U\cap V,x_0)\) factors uniquely through the free product modulo \(N\)). A

Result

\(X = U\cup V,\ \ U,V,U\cap V\ \text{open, path-connected}\ \implies\ \pi_1(X,x_0) \cong \pi_1(U,x_0) \ast_{\pi_1(U\cap V,x_0)} \pi_1(V,x_0)\)

Reading. Any loop in \(X\) can be cut into finitely many pieces each living in \(U\) or in \(V\), so every element of \(\pi_1(X)\) comes from \(\pi_1(U)\) and \(\pi_1(V)\); and the only relations forced between these pieces are the ones you'd expect — that a loop sitting in the overlap represents "the same" element whether you view it as living in \(U\) or in \(V\).

Scope. Applies to any space admitting an open cover by two path-connected sets with path-connected intersection sharing a common point; extends (via the fundamental groupoid) to covers by any number of open sets, or to \(U\cap V\) with several path components, at the cost of replacing the group pushout with a groupoid pushout/colimit. Does not by itself compute higher homotopy groups, which do not satisfy an analogous van Kampen theorem in general (though there are versions for \(n\)-connected covers and simplicial techniques).

Corollaries & converses

If \(U\cap V\) is simply connected, the amalgamation collapses to a free product: \(\pi_1(X,x_0) \cong \pi_1(U,x_0) \ast \pi_1(V,x_0)\) (no relations imposed, since \(N\) is generated by the trivial group's image).
If in addition \(U\) or \(V\) is simply connected, \(\pi_1(X,x_0)\) is a quotient of the other factor by the normal closure of \((i)_\ast(\pi_1(U\cap V))\); if both \(U,V,U\cap V\) are simply connected then \(X\) is simply connected.
Attaching an \(n\)-cell (\(n\geq 3\)) to a space does not change \(\pi_1\) (take \(V\) a small neighbourhood of the cell, deformation retracting to the cell so simply connected once \(n\geq 3\) makes \(U\cap V \simeq S^{n-1}\) simply connected too); attaching a \(2\)-cell along a loop \(\gamma\) kills \(\gamma\) as a normal generator — this is exactly the group-presentation-from-CW-structure recipe.
Converse: false as stated. Knowing \(\pi_1(X)\) is isomorphic to some amalgamated product \(A\ast_C B\) does not imply \(X\) splits as such a union with those groups as the pieces' fundamental groups — the algebraic decomposition of a group as an amalgam is far from unique (e.g. \(\mathbb{Z}\ast_{2\mathbb{Z}}\mathbb{Z}\) and other coincidences), and abstract group-theoretic amalgam decompositions carry no topological meaning without an accompanying open cover.

Fails without

Openness dropped: cover \(S^1\) by two closed semicircular arcs \(A,B\) meeting only at their two shared endpoints. Both are contractible and \(A\cap B\) is two points (not path-connected, and not open), so no Lebesgue-number subdivision argument applies; there is no meaningful pushout statement to write down, and certainly \(\pi_1(S^1)=\mathbb{Z}\) is not recovered from any naive attempt.
\(U\cap V\) path-connected dropped: \(S^1 = U\cup V\) with \(U,V\) open arcs each slightly more than a semicircle, so \(U\cap V\) is two disjoint arcs. Here \(U,V\) are contractible so \(\pi_1(U)=\pi_1(V)=1\); the (incorrect) formula \(1\ast_{\pi_1(U\cap V)}1\) would give the trivial group, but \(\pi_1(S^1)\cong\mathbb{Z}\neq 1\). The extra generator "\(\mathbb{Z}\)" comes precisely from having two disjoint copies of \(U\cap V\) to cross, information the group-pushout statement cannot see; the correct fix uses the fundamental groupoid on two basepoints, one in each component of \(U\cap V\).
\(U,V\) path-connected dropped: let \(X=\mathbb{R}^2\), \(U = X\setminus\{(1,0)\}\), and pick \(V\) a small disc around \((1,0)\) union a disjoint small disc far away, so \(V\) is disconnected while \(U\cap V\) is only the first disc, path-connected, but \(V\) itself is not; the induced maps \((j_V)_\ast\) become ill-defined on the disconnected component not containing \(x_0\), and any loop passing through that stray piece of \(V\) is invisible to \(\pi_1(V,x_0)\), so \(\Phi\) can fail to be surjective onto \(\pi_1(X,x_0)\) once such a piece is attached in a way that actually threads through both.

Common errors

Forgetting to check \(U\cap V\) is path-connected before quoting the theorem — the single most common error, since many natural covers (e.g. by two hemispheres of \(S^1\)) fail exactly this hypothesis.
Amalgamating over the wrong group: writing \(\pi_1(X)\cong\pi_1(U)\ast\pi_1(V)\) (plain free product) when \(\pi_1(U\cap V)\) is nontrivial and its images in \(U\), \(V\) are not both trivial — the relations \(i_U{}_\ast(\gamma)=i_V{}_\ast(\gamma)\) are then essential and dropping them changes the answer, e.g. for the torus decomposition this error would give a nonabelian free group instead of \(\mathbb{Z}^2\).
Confusing "\(U\cap V\) simply connected" with "\(U\cap V\) is a point" — simple connectivity is all that is needed for the amalgam to reduce to a free product, no contractibility of the overlap to a single point is required.
Using closed sets instead of open sets in the cover (e.g. taking \(U,V\) to be the closed upper/lower hemispheres of \(S^n\)) — the theorem as stated genuinely requires openness for the subdivision argument; a common patch (replacing closed hemispheres by slightly enlarged open neighbourhoods that deformation retract onto them) is needed and often silently skipped.
Applying the two-set theorem directly to a cover by more than two sets without correctly iterating it pairwise (or invoking the multi-set groupoid version), which can silently drop cross-terms between non-adjacent pieces of the cover.

Discussion

The theorem is named for Herbert Seifert and Egbert van Kampen, who independently proved versions of it in the 1930s (Seifert 1931, van Kampen 1933), in the broader project of using combinatorial/simplicial decompositions to compute topological invariants; van Kampen's original proof already handled the amalgamated-product statement essentially as given here, phrased in terms of generators and relations rather than the modern categorical pushout language.

Its structural role becomes clearer once one recognises \(\pi_1\) as a functor from pointed spaces to groups that does not preserve colimits on the nose — a union is a pushout of spaces (\(U \leftarrow U\cap V \to V\), with \(X\) the pushout), and van Kampen says this particular functor does take that pushout of spaces to the pushout of groups, provided the gluing region is connected enough to see. This is the same shape of theorem as Mayer–Vietoris in homology, which computes \(H_\ast(X)\) from a long exact sequence built out of \(H_\ast(U), H_\ast(V), H_\ast(U\cap V)\); van Kampen is the non-abelian, \(\pi_1\)-level analogue, and the reason it takes a pushout rather than an exact sequence is precisely that groups (unlike abelian groups) do not have a well-behaved notion of "difference" of two maps.

A cleaner modern proof route replaces the explicit grid-combinatorics of Steps 5–9 with the fundamental groupoid \(\Pi_1(X)\) on a suitable set of basepoints and a categorical van Kampen theorem for groupoids (due to Ronald Brown), which is a colimit statement requiring no connectivity hypothesis on \(U\cap V\) at all — the classical group-level theorem is recovered afterward by specialising to a single basepoint once \(U\cap V\) happens to be connected. This explains structurally why the group-level statement is so brittle to disconnected overlaps: the "right" object to take a colimit of was always the groupoid, and passing to \(\pi_1\) at one point only works when there is nothing lost in doing so.

Common misconception: that the theorem says \(\pi_1(U\cup V)\) is "built from" \(\pi_1(U)\) and \(\pi_1(V)\) alone, with \(U\cap V\) merely a technical convenience. In fact \(\pi_1(U\cap V)\) plays an equally load-bearing role: it is not the group \(\pi_1(U\cap V)\) itself that matters but the pair of homomorphisms out of it, and changing those homomorphisms (while keeping all three groups fixed) generally changes \(\pi_1(X)\) — the amalgam depends on the maps, not merely on the three groups in isolation.

Worked examples

Compute \(\pi_1(S^1\vee S^1,x_0)\), the wedge of two circles at their common point \(x_0\).

Setup: let \(A, B\) be the two circles. Take \(U = A \cup (\text{small open segment of }B\text{ near }x_0)\), and \(V = B \cup (\text{small open segment of }A\text{ near }x_0)\), each open in \(X\), each deformation-retracting onto its circle so \(\pi_1(U,x_0)\cong\mathbb{Z}\), \(\pi_1(V,x_0)\cong\mathbb{Z}\). A

\(U\cap V\) is the union of the two small open segments, an open star-shaped neighbourhood of \(x_0\) — contractible, hence simply connected and (as required) path-connected.

Direct check of the construction: the two segments meet only at \(x_0\) and each is contractible to \(x_0\) within itself, and a small enough star neighbourhood retracts to the point. A

By the theorem, \(\pi_1(S^1\vee S^1,x_0) \cong \pi_1(U,x_0)\ast_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0) = \mathbb{Z}\ast_1\mathbb{Z} = \mathbb{Z}\ast\mathbb{Z}\), the free group on two generators.

Since \(\pi_1(U\cap V,x_0)\) is trivial, \(N\) is generated by images of the identity element only, so \(N=\{1\}\) and the amalgam is the ordinary free product \(\mathbb{Z}\ast\mathbb{Z} = F_2\) (Corollary 1 above). B

\(\pi_1(S^1\vee S^1) \cong F_2 = \langle a,b \mid\, \rangle\), the free group on two generators, one per circle.

Reading. Loops in a figure-eight are exactly words in the two loop-generators \(a\) (around the first circle) and \(b\) (around the second), with no relations forcing any word to simplify.

Compute \(\pi_1(S^n,x_0)\) for \(n\geq 2\).

Setup: let \(p,q\in S^n\) be the north and south poles, and take \(U = S^n\setminus\{q\}\), \(V=S^n\setminus\{p\}\), each open (complement of a point). Stereographic projection gives homeomorphisms \(U\cong\mathbb{R}^n\), \(V\cong\mathbb{R}^n\), both contractible, so \(\pi_1(U,x_0)=\pi_1(V,x_0)=1\) for any basepoint \(x_0\in U\cap V\). A

\(U\cap V = S^n\setminus\{p,q\} \cong S^{n-1}\times\mathbb{R}\), which is path-connected for \(n\geq 2\) (since \(S^{n-1}\) is path-connected for \(n-1\geq 1\), i.e. \(n\geq 2\)).

Removing two points from \(S^n\) and using the standard "cylinder" homeomorphism \(S^n\setminus\{p,q\}\cong S^{n-1}\times\mathbb{R}\) (radial projection away from the polar axis); path-connectedness of a product with a path-connected factor \(S^{n-1}\) (\(n\geq 2\)) is immediate. A

By the theorem, \(\pi_1(S^n,x_0) \cong \pi_1(U,x_0)\ast_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0) = 1\ast_{\pi_1(S^{n-1}\times\mathbb{R})}1 = 1\) for every \(n\geq 2\), regardless of what \(\pi_1(S^{n-1}\times\mathbb{R})\) actually is.

An amalgamated product of two trivial groups is trivial no matter what group they are amalgamated over: any homomorphism out of the trivial group has trivial image, so \(N\) is trivially all of the (trivial) free product, and the quotient is trivial. Note this argument does not need to know \(\pi_1(S^{n-1})\) at all — it would work even if \(U\cap V\) were, say, not simply connected. B

\(\pi_1(S^n) = 1\) for all \(n\geq 2\) (and \(S^n\) is simply connected).

Reading. The sphere splits into two contractible caps overlapping in a connected annular region; since both caps individually kill all loops, no loop anywhere in \(S^n\) can survive, irrespective of the overlap's own fundamental group.

Problems

Use the standard CW/polygon decomposition of the torus \(T^2\) (a square with edges identified \(aba^{-1}b^{-1}\)) to compute \(\pi_1(T^2)\) via the Seifert–van Kampen theorem, taking \(U\) a small open disc around the centre of the square and \(V\) the complement of the centre point (deformation retracting onto the wedge of two circles \(a,b\) from the boundary identification).
Solution
Let \(X=T^2\) with its usual identification of a square via \(aba^{-1}b^{-1}\); let \(x_0\) be the (single) vertex, which is where all four corners are glued. Take \(U\) a small open disc around the centre of the square (not touching the boundary), so \(U\) is contractible: \(\pi_1(U)=1\). Take \(V = X\setminus\{\text{centre}\}\), which deformation retracts onto the boundary edge-word, i.e. onto \(S^1\vee S^1\) with generators \(a,b\); so \(\pi_1(V,x_0)\cong F_2=\langle a,b\rangle\) (basepoint moved to the wedge point via the retraction). \(U\cap V\) is an annulus (disc minus its centre point... more precisely disc minus centre, deformation retracting onto a small circle around the centre), which is path-connected with \(\pi_1(U\cap V)\cong\mathbb{Z}\), generated by a loop \(\delta\) that traces once around the centre. Tracing \(\delta\) through the retraction of \(V\) onto the boundary word shows \(i_V{}_\ast(\delta)\) is exactly the boundary edge-path \(aba^{-1}b^{-1}\) (reading the square's boundary once around), while \(i_U{}_\ast(\delta)=1\) since \(\pi_1(U)\) is trivial. By the theorem, \(\pi_1(T^2)\cong \pi_1(U)\ast_{\pi_1(U\cap V)}\pi_1(V) = 1\ast_{\mathbb{Z}}F_2\), which forces the single relation \(i_V{}_\ast(\delta)=i_U{}_\ast(\delta)=1\), i.e. \(aba^{-1}b^{-1}=1\). So \(\pi_1(T^2)\cong\langle a,b\mid aba^{-1}b^{-1}\rangle\cong\mathbb{Z}\times\mathbb{Z}=\mathbb{Z}^2\).
Compute \(\pi_1(\mathbb{RP}^2,x_0)\) using the polygon decomposition of \(\mathbb{RP}^2\) as a square with edge word \(aa\) (i.e. \(a^2\)), by the same \(U\)/\(V\) split as in Problem 1.
Solution
As in Problem 1, take \(U\) a small disc around the centre (\(\pi_1(U)=1\)) and \(V\) the complement of the centre, retracting onto the boundary circle \(S^1\) with a single generator \(a\) (the square has only one edge-pair, glued as \(a a\)), so \(\pi_1(V,x_0)\cong\langle a\rangle\cong\mathbb{Z}\) freely (before imposing the relation from the overlap). The overlap generator \(\delta\) again maps under \(i_V{}_\ast\) to the boundary word read once around, which is \(a\cdot a = a^2\), while \(i_U{}_\ast(\delta)=1\). So the amalgam \(1\ast_{\mathbb{Z}}\mathbb{Z}\) imposes \(a^2=1\), giving \(\pi_1(\mathbb{RP}^2)\cong\langle a\mid a^2\rangle\cong\mathbb{Z}/2\mathbb{Z}\).
(Disconnected overlap.) Cover \(S^1\) by two open arcs \(U,V\) each covering more than a semicircle, so that \(U\cap V\) has two path components \(P_1\ni x_0\) and \(P_2\), with a point \(x_1\in P_2\). Explain precisely why the group-level Seifert–van Kampen theorem as stated cannot be applied directly to compute \(\pi_1(S^1,x_0)\), and use the groupoid idea (tracking a path \(\eta\) from \(x_0\) to \(x_1\) through \(U\), and separately through \(V\)) to recover \(\pi_1(S^1,x_0)\cong\mathbb{Z}\) informally.
Solution
The theorem as stated requires \(U\cap V\) path-connected; here it has two components \(P_1,P_2\), so the hypothesis fails and the pushout-of-groups conclusion is simply not asserted by the theorem (applying it anyway, as noted in "Fails without", would wrongly predict the trivial group since \(\pi_1(U)=\pi_1(V)=1\)). Informally: choose a path \(\eta_U\) from \(x_0\) to \(x_1\) inside \(U\), and a path \(\eta_V\) from \(x_0\) to \(x_1\) inside \(V\) (both exist since \(U,V\) are individually path-connected arcs). The loop \(\eta_U\cdot\eta_V^{-1}\) is a genuine loop at \(x_0\) in \(X=S^1\), and it is exactly one generator that neither \(\pi_1(U,x_0)\) nor \(\pi_1(V,x_0)\) alone can see, because it is built from crossing the overlap through two different components. One shows (this is the content of the groupoid van Kampen theorem) that \(\pi_1(S^1,x_0)\) is generated by this single class \([\eta_U\eta_V^{-1}]\) with no relations, i.e. it is infinite cyclic; this matches \(\pi_1(S^1)\cong\mathbb{Z}\), with the extra generator arising precisely from the second overlap component that the naive group-only statement discards.
Prove directly from the universal property of the pushout (not the explicit grid proof) that if \(\pi_1(U,x_0)=\pi_1(V,x_0)=1\) then \(\pi_1(U,x_0)\ast_{\pi_1(U\cap V,x_0)}\pi_1(V,x_0)=1\), for an arbitrary group \(\pi_1(U\cap V,x_0)\) and arbitrary homomorphisms into the trivial groups.
Solution
The pushout \(P\) of \(1 \xleftarrow{} C \xrightarrow{} 1\) (where \(C=\pi_1(U\cap V,x_0)\) and both maps are necessarily the unique homomorphism to the trivial group) is, by the universal property, the group \(P\) equipped with homomorphisms \(1\to P\), \(1\to P\) making the square commute, such that any other group \(Q\) with compatible maps from the two copies of \(1\) receives a unique map \(P\to Q\) making everything commute. But the trivial group \(1\) itself, with its unique maps, satisfies this universal property vacuously: for any \(Q\) there is a unique homomorphism \(1\to Q\) (sending the identity to the identity), so \(1\) already has the required unique factorisation property. By uniqueness of objects representing a universal property (up to unique isomorphism), \(P\cong 1\). Concretely this matches Step 3 of Worked Example 2: the amalgam of two trivial groups over anything is trivial.
Let \(X\) be the union of two circles glued at two distinct points (a "theta graph" \(\theta\), i.e. two vertices joined by three edges). Decompose \(X\) into two open sets \(U,V\) suitable for the Seifert–van Kampen theorem and compute \(\pi_1(X,x_0)\).
Solution
Label the two vertices \(p,q\) and the three edges \(e_1,e_2,e_3\), each an arc from \(p\) to \(q\); pick the basepoint \(x_0=p\). Let \(U\) be a small open neighbourhood of the union \(e_1\cup e_2\) (thickened slightly past \(q\) so it stays open), which deformation retracts onto the circle \(e_1\cup e_2\) (a loop through \(p\) and \(q\)); so \(\pi_1(U,x_0)\cong\mathbb{Z}\), generated by \(c_{12}=e_1 e_2^{-1}\) (go along \(e_1\) from \(p\) to \(q\), back along \(e_2\)). Let \(V\) be a small open neighbourhood of \(e_2\cup e_3\), similarly retracting onto a circle with \(\pi_1(V,x_0)\cong\mathbb{Z}\), generated by \(c_{23}=e_2e_3^{-1}\). Arrange the neighbourhoods so \(U\cap V\) is a small open neighbourhood of \(e_2\) together with small neighbourhoods of \(p,q\), which deformation retracts onto \(e_2\), an arc — contractible, so \(\pi_1(U\cap V,x_0)=1\). By the theorem (trivial overlap group, so free product as in Corollary 1), \(\pi_1(X,x_0)\cong\mathbb{Z}\ast\mathbb{Z}=F_2\), the free group on two generators \(c_{12},c_{23}\) — consistent with \(X\) being homotopy equivalent to a wedge of two circles (a graph with 2 vertices and 3 edges has Euler characteristic \(2-3=-1\), so it is homotopy equivalent to a wedge of \(1-\chi=2\) circles).