Wedges of Surfaces and Free Product Presentations

Why a Wedge Produces a Free Product

When two connected surfaces are joined at one chosen point, a loop can travel through either surface and return to the joining point. The two surfaces share that point, but their other points and loops are not identified. The Seifert–van Kampen theorem captures this by making the fundamental group of the wedge a free product: loops from each surface retain their own relations, with no additional relations imposed between the two sets of loops.

All surfaces below have their usual topology and are based at specified points. For based spaces \((X,x_0)\) and \((Y,y_0)\), their wedge \(X\vee Y\) is the quotient of the disjoint union \(X\bigsqcup Y\) in which \(x_0\) and \(y_0\) are identified to one point. The wedge has the quotient topology: a set is open exactly when its preimage in each summand is open. The identified point is the basepoint of \(X\vee Y\).

We use the standard presentations for closed connected surfaces. The sphere \(S^2\) has trivial fundamental group. The closed orientable surface \(\Sigma_g\) of genus \(g\geq1\) has presentation with generators \(a_1,b_1,\ldots,a_g,b_g\) and the single relator \([a_1,b_1]\cdots[a_g,b_g]\), where \([a,b]=aba^{-1}b^{-1}\). The closed nonorientable surface \(N_k\) of genus \(k\geq1\) has generators \(c_1,\ldots,c_k\) and relator \(c_1^2\cdots c_k^2\). These surface presentations are standard results established by the polygon and van Kampen computations for surfaces.

The Wedge Theorem

Proof. Choose open disk neighborhoods \(D_x\subseteq X\) of \(x_0\) and \(D_y\subseteq Y\) of \(y_0\). In the wedge let \(U=X\cup D_y\) and \(V=D_x\cup Y\), where each union uses the identified basepoint. These sets are open: their preimages in the two summands are respectively \((X,D_y)\) and \((D_x,Y)\), all of which are open. They cover \(X\vee Y\), and \(U\cap V=D_x\vee D_y\).

The radial contraction of \(D_y\) to \(y_0\), leaving \(X\) fixed, descends through the quotient and gives a deformation retraction of \(U\) onto \(X\). The radial contraction of \(D_x\) gives a deformation retraction of \(V\) onto \(Y\). Contracting both disks to their centers contracts \(D_x\vee D_y\) to the wedge point, so \(U\cap V\) is path-connected and simply connected. The Seifert–van Kampen theorem for this open cover therefore identifies \(\pi_1(X\vee Y)\) with the free product of the groups of \(U\) and \(V\), since the intersection has trivial fundamental group. The deformation retractions identify those groups with \(\pi_1(X,x_0)\) and \(\pi_1(Y,y_0)\). This proves the claimed isomorphism.

For the presentation statement, take disjoint generator sets \(S_X,S_Y\) and relator sets \(R_X,R_Y\) presenting the two groups. The union presentation has generators \(S_X\cup S_Y\) and relators \(R_X\cup R_Y\). The algebraic reason it presents a free product is proved below. \(\square\)

The argument also applies to any finite wedge of path-connected closed surfaces: apply the two-summand result repeatedly. If a summand is \(S^2\), its group is trivial and contributes no generators or relators. The theorem concerns based wedges; it does not say that arbitrary surfaces joined along a circle have free-product groups.

Why the Union of Presentations Works

Proof. Let \(P=\langle S\cup T\mid R\cup Q\rangle\). The generator assignment from \(S\) to its elements in \(G\) and from \(T\) to its elements in \(H\) respects every relator in \(R\cup Q\). It therefore gives a homomorphism \(P\to G*H\). In the other direction, the inclusions of the generators into \(P\) respect \(R\) and \(Q\), so they induce homomorphisms \(G\to P\) and \(H\to P\). By the defining universal property of the free product, these induce a homomorphism \(G*H\to P\).

The composite \(P\to G*H\to P\) fixes every generator in \(S\cup T\), so it is the identity on \(P\). The composite \(G*H\to P\to G*H\) restricts to the identity on both \(G\) and \(H\); the universal property says it is the identity on \(G*H\). Thus the homomorphisms are inverse isomorphisms, proving the presentation claim. \(\square\)

This proposition is what makes the calculation systematic: retain each summand’s relators, and do not add a relator involving generators from different summands. The free product allows arbitrary alternating words from the two groups unless their own relators reduce them.

A Common Pitfall: Combining Relators into One

A tempting but incorrect shortcut is to multiply relators from the two summands and keep only that product as a relation. For the torus and projective-plane example, replacing the two relators by the single relation \([a,b]c^2=1\) does not impose each original relation separately. Here is a concrete check. In the group of symmetries of a square, write \(r\) for a rotation by a quarter-turn and \(s\) for a reflection. They satisfy \(r^4=s^2=1\) and \(srs=r^{-1}\). Assign \(a=s\), \(b=r\), and \(c=r\). Then

So this assignment satisfies the proposed single product relation, but \([a,b]=r^2\ne1\) and \(c^2=r^2\ne1\). It does not satisfy the torus and projective-plane relators separately. The actual wedge presentation must retain both relators. The free-product presentation is built from the union of the relator sets, not from an arbitrary product of relators.

The calculation matters because it preserves information about how loops in the separate surfaces interact. A free product is not generally a direct product: a direct product would additionally require every element from one factor to commute with every element from the other. In a wedge, the shared basepoint alone does not impose those commutation relations.