From Two Surfaces to One
A polygon diagram describes a surface by pairing edges of a polygon. A connected sum gives a geometric way to combine two such descriptions: remove a small disk from each surface, then join the newly exposed boundary circles. The operation keeps the two surfaces’ edge-identification data while bringing their models into one connected surface.
Here a closed surface means a compact Hausdorff 2-manifold without boundary. A coordinate disk is a subset of the surface that is the image of a closed disk under a coordinate chart, with its boundary corresponding to the boundary circle of that disk. Removing the interior of a coordinate disk leaves a surface with one boundary circle.
The boundary circles are not left as boundary components: they are identified point by point. Thus the construction joins the two remaining surfaces along their entire boundary circles. A polygon boundary word records the paired edges before the operation; the connected-sum diagram will combine the two words using distinct labels for the two surfaces’ edge classes.
Why the Gluing Is Still a Surface
Proof. Write \(M_0=M\setminus\operatorname{int}D_M\) and \(N_0=N\setminus\operatorname{int}D_N\). Each is compact and has one boundary circle. At points away from the identified circles, a neighborhood in the quotient is just a neighborhood in \(M_0\) or \(N_0\), so it is an open disk. Each boundary circle has a collar: a neighborhood homeomorphic to \([0,1)\times S^1\), with the boundary corresponding to \(\{0\}\times S^1\). After the circles are identified, the two collars join to form a neighborhood homeomorphic to \((-1,1)\times S^1\). Points on the join therefore also have disk neighborhoods. There is no boundary left.
The disjoint union \(M_0\sqcup N_0\) is compact, so its quotient is compact. It is Hausdorff as well: the points identified form a closed subset, and the equivalence relation consists of the diagonal together with the graph of \(h\) and its inverse. This is a closed relation on a compact Hausdorff space; the quotient by such a relation is Hausdorff. For connectedness, each of \(M_0,N_0\) is connected. Indeed, a component of \(M_0\) not meeting its boundary would be both open and closed in \(M\), contradicting connectedness of \(M\); the same argument applies to \(N_0\). Their images in the quotient are connected and meet along the identified circle, so their union is connected. Thus the quotient is a compact connected 2-manifold without boundary. \(\square\)
In a polygon diagram, each pair of identified edges has a label. A letter and its inverse, such as \(a\) and \(a^{-1}\), indicate that the two edges are glued with opposite directions along the polygon boundary. Two occurrences of \(a\), written \(aa\), indicate a pairing with matching boundary directions. The direction convention is part of the diagram, not an optional decoration.
Proof. A disk chosen inside a polygon face does not meet any of the paired polygon edges, so removing its interior leaves all edge pairings unchanged. In each punctured polygon, cut along an arc joining the new circular boundary to the polygon boundary. After the two circular boundaries are glued, the two cut arcs join. Opening the model along this joined arc gives one polygon: following its boundary first traces the old edge sequence of the first diagram and then the old edge sequence of the second. The two sides created by the cut arc are identified with each other and are not additional labelled edges. Since the original pairings are unchanged and the two diagrams had separate edge classes, their labels must remain distinct. This gives the concatenated word. \(\square\)
Connected Sums in Polygon Words
Worked Example: The Sum of Two Tori
A torus has the standard square boundary word \(aba^{-1}b^{-1}\). Take two copies, use \(c,d\) for the second copy’s edge classes, and apply the polygon construction rule. The connected-sum word is
To check the diagram’s vertex identifications, number the nine consecutive boundary vertices \(v_0,\ldots,v_8\). The eight edges run in order from \(v_0\) to \(v_8\). Pairing \(a\) at \(v_0v_1\) with \(a^{-1}\) at \(v_2v_3\) identifies \(v_0\sim v_3\) and \(v_1\sim v_2\). Pairing \(b\) at \(v_1v_2\) with \(b^{-1}\) at \(v_3v_4\) identifies \(v_1\sim v_4\) and \(v_2\sim v_3\). Together these make \(v_0,\ldots,v_4\) one class.
For the second block, \(c\) at \(v_4v_5\) pairs with \(c^{-1}\) at \(v_6v_7\), giving \(v_4\sim v_7\) and \(v_5\sim v_6\). The \(d\) edge at \(v_5v_6\) pairs with \(d^{-1}\) at \(v_7v_8\), giving \(v_5\sim v_8\) and \(v_6\sim v_7\). The first block already identifies \(v_4\) with \(v_0,\ldots,v_3\); the second block’s relations then identify \(v_5,v_6,v_7,v_8\) with that same class. Hence all nine vertices belong to one equivalence class.
Check: the word contains eight edge occurrences, forming the four pairs \(a,b,c,d\), and the vertex relations above account for every one of \(v_0,\ldots,v_8\). No vertex is left out of the single class.
Worked Example: The Sum of Two Projective Planes
The standard paired-polygon model of the projective plane has boundary word \(aa\): the two occurrences of \(a\) are identified with matching boundary directions. For a second projective plane, use the new label \(b\). Concatenating the words gives
Number the five consecutive boundary vertices \(v_0,\ldots,v_4\). The two \(a\)-edges run from \(v_0\) to \(v_1\) and from \(v_1\) to \(v_2\). Matching-direction identification gives \(v_0\sim v_1\) and \(v_1\sim v_2\). The two \(b\)-edges run from \(v_2\) to \(v_3\) and from \(v_3\) to \(v_4\); their identification gives \(v_2\sim v_3\) and \(v_3\sim v_4\). By transitivity, \(v_0\sim v_1\sim v_2\sim v_3\sim v_4\), so the quotient has one vertex class.
Check: there are four edge occurrences, paired as the two \(a\)-edges and the two \(b\)-edges. The listed endpoint identifications connect all five boundary vertices, including the first and last vertices of the polygon.
Worked Example: A Torus Summed with a Projective Plane
Use \(aba^{-1}b^{-1}\) for the torus and \(cc\) for the projective plane, choosing \(c\) distinct from \(a\) and \(b\). The connected-sum word is
There are six edges and seven consecutive boundary vertices \(v_0,\ldots,v_6\). The torus pairings give \(v_0\sim v_3\), \(v_1\sim v_2\), \(v_1\sim v_4\), and \(v_2\sim v_3\). These relations identify \(v_0,\ldots,v_4\). The two \(c\)-edges run from \(v_4\) to \(v_5\) and from \(v_5\) to \(v_6\). Since they have matching directions, they give \(v_4\sim v_5\) and \(v_5\sim v_6\). Thus all seven vertices are in one class.
Check: the three edge pairs are \(a\) with \(a^{-1}\), \(b\) with \(b^{-1}\), and \(c\) with \(c\). The vertex relations connect the final two vertices to the class already formed by \(v_0,\ldots,v_4\).
A Common Labeling Error
Common pitfall. When forming a connected sum, it is wrong to reuse the same labels for unrelated edge classes in the two original diagrams. For example, writing \(aba^{-1}b^{-1}ab\) for a torus summed with a projective plane treats the projective plane’s two edges as if they were already the torus’s \(a\)- and \(b\)-edges. That word has no separate pair consisting of two occurrences of a new projective-plane label, so it does not record the two original edge pairings.
The correction is to introduce a fresh label for the projective-plane pair. Its word is \(cc\), so the combined word is \(aba^{-1}b^{-1}cc\). The six occurrences then divide into the torus’s two inverse pairs and the projective plane’s matching-direction pair. Keeping edge classes distinct preserves the original identifications; it does not mean that the two surfaces remain separate, since the connected sum still joins their punctured pieces along a circle.
The polygon word is a compact record of the construction, but the construction itself is the disk removal and circle gluing. In particular, the direction of each edge pairing remains essential: changing \(cc\) to \(cc^{-1}\), for instance, changes the prescribed gluing. The examples show how to form a combined word and then check its vertex classes directly from the endpoint identifications.
Check Your Understanding
For each question, keep the edge labels of the two original diagrams distinct and use the stated direction convention.
- Write the connected-sum boundary word for a torus and a projective plane, and state which edge occurrences are paired.
- For the word \(aabb\), list the vertex identifications contributed by the two edge pairs.
- Why must an oriented connected-sum construction use a boundary map that reverses the induced boundary orientations?
- In the torus-plus-projective-plane word, which identifications show that the final two boundary vertices join the torus’s vertex class?
- Explain why a word that reuses the torus labels for the projective-plane edges fails to record the original two edge pairings.