The Genus Two Surface Group

From an Octagon to a Group Presentation

A closed orientable surface of genus two can be assembled from two handles. Its fundamental group records loops around both handles and the way the surface closes up. A useful model makes this information explicit: identify the sides of an octagon in pairs, then regard the octagon as a disk attached to a graph. The Seifert–van Kampen theorem converts that attachment into a relation.

Throughout, fundamental groups are based at a specified point, and multiplication of loops means traversing the left-hand loop first and then the right-hand loop. For group elements \(x,y\), write \([x,y]=xyx^{-1}y^{-1}\). A presentation \(\langle x_1,\ldots,x_n\mid r\rangle\) means the free group on the listed generators, divided by the normal subgroup generated by the relator \(r\).

This model also displays the surface as a disk attached to that four-loop graph. The first four sides describe one handle and the last four describe the other: grouping the octagon boundary into these two blocks gives the two commutator loops. Equivalently, the surface can be viewed as two once-punctured tori joined along their boundary circles. We will use the disk-and-graph description because it keeps the attaching word and the basepoint visible.

Reading the Attaching Word

Proof. Start at the unique vertex and traverse the boundary once in its specified order. The first four edges contribute \(a b a^{-1} b^{-1}\), which is \([a,b]\) by the definition of commutator. The last four edges contribute \(c d c^{-1} d^{-1}\), which is \([c,d]\). Concatenating the two parts gives \(a b a^{-1} b^{-1} c d c^{-1} d^{-1}=[a,b][c,d]\). The endpoint is the starting vertex because all polygon vertices are identified. \(\square\)

The order of the factors matters: a fundamental group need not be abelian, so in general \([a,b][c,d]\) is not interchangeable with \([c,d][a,b]\). The polygon specifies the order. Reversing the direction around the entire boundary replaces the word by its inverse, which imposes the same normal-closure relation.

Applying Seifert–van Kampen

In this form, the theorem says that attaching the disk makes its boundary loop null-homotopic, along with every conjugate of that loop. It does not say that each piece of the boundary word becomes trivial separately.

Proof. Let \(G\) be the four-loop graph in the octagon model, and let \(X\) be the closed surface obtained by attaching the octagonal disk. The wedge-of-circles computation gives \(\pi_1(G)=F(a,b,c,d)\), the free group on the four edge loops; this is the standard result for wedges of circles established earlier. To apply Seifert–van Kampen, take a collar of the attaching circle in the disk. Let \(U\) consist of \(G\) together with the part of the disk in this collar, and let \(V\) be the disk with a smaller collar removed from its boundary, chosen so that the collar regions overlap. These are open sets in the attached space, \(U\cup V=X\), and \(U\cap V\) is an annulus. Radial contraction retracts \(U\) to \(G\); \(V\) is contractible; and \(U\cap V\) is path connected and deformation retracts to a circle. Choose the basepoint on the annulus above the polygon vertex, so its image in \(G\) is the graph vertex. The generator of \(\pi_1(U\cap V)\cong\mathbb Z\) maps into \(\pi_1(U)\) as the attaching loop \([a,b][c,d]\), by the proposition, and maps to the identity in \(\pi_1(V)\), since \(V\) is contractible. The Seifert–van Kampen theorem therefore gives the quotient of \(F(a,b,c,d)\) by the normal closure of \([a,b][c,d]\). A quotient by that normal closure is exactly the group specified by the displayed presentation. \(\square\)

The same computation explains the two-handle description. One punctured torus contributes the loops \(a,b\), with boundary represented by \([a,b]\); the other contributes \(c,d\), with boundary represented by \([c,d]\). When the boundary circles are joined to close the surface, the oriented boundary word around the full polygon is their product. The disk attachment imposes that product as a single relation.

Choices of Direction and Starting Point

A polygon can be read from a different starting vertex, or its boundary can be traversed in reverse. These choices can change the written word without changing the relation imposed by the disk. The relevant group-theoretic fact is that conjugate relators generate the same normal subgroup, and a relator and its inverse generate the same normal subgroup.

Proof. The normal closure of \(w\) is the smallest normal subgroup containing \(w\). Any subgroup containing \(w\) contains \(w^{-1}\); and a normal subgroup containing \(w\) contains \(g^{-1}wg\) for every \(g\). Thus the normal closure of \(w\) contains \(w^{-1}\) and \(g^{-1}wg\). Conversely, \(w=(w^{-1})^{-1}\), and \(w=g(g^{-1}wg)g^{-1}\); hence the normal closure of either \(w^{-1}\) or \(g^{-1}wg\) contains \(w\). Each of the three normal closures therefore contains the other two, so they are equal. \(\square\)

A Common Mistake: Splitting One Relation into Two

A tempting but incorrect reading of \([a,b][c,d]=1\) is that it requires \([a,b]=1\) and \([c,d]=1\) separately. A product can be the identity because its factors are inverses, even when neither factor is the identity.

Concrete correction. In the free group on \(x,y\), set \(a=x\), \(b=y\), \(c=y\), and \(d=x\). Then \([a,b]=xyx^{-1}y^{-1}\), while \([c,d]=yxy^{-1}x^{-1}=[a,b]^{-1}\). Consequently \([a,b][c,d]=1\), although \([a,b]\) is not the identity word in the free group. This assignment satisfies the single product relation without making either commutator trivial. The genus-two presentation therefore has one relation joining the two handles, not two separate commutator relations.

The important bookkeeping is: the four edge loops give four generators; the disk boundary gives one word; and attaching the disk kills that word and its conjugates. The resulting presentation records exactly the identification required to close the genus-two surface.