An n-dimensional manifold is a Hausdorff space that is locally isomorphic to Euclidean n-space E(n).
In the two-dimensional case, the basic examples are the sphere, let us call it M(0), the torus, let us call it M(1), and the real projective plane, let us call it N(1). Arbitrary examples are constructed by glueing. We use the symbol # again. Given two n-manifolds X, Y, and two homeomorphisms f : B -> X and g : B -> Y of the closed unit sphere into these spaces, let X#Y denote the space obtained from the disjoint union of X and Y by removing the images of the open unit sphere, and identifying images of the boundary points in X with corresponding images of boundary points in Y. Clearly, X#M(0) = X.
Let M(g) = M(g-1)#M(1) be the sphere with g handles. Let N(h) = N(h-1)#N(1) be the sphere with h crosscaps.
Theorem Every compact connected 2-manifold is homeomorphic to one of the surfaces M(g) (g > 0) or N(h) (h > 1).
In the proof we shall need the concept triangulation of a surface. This is a covering of the space with finitely many triangles, homeomorphic images of an ordinary plane triangle, where two triangles either are disjoint, or have a point or an edge in common, each edge is in precisely two triangles, and the residue at a point is a polygon.
First of all, these surfaces are all distinct, since they have
distinct fundamental groups, see below. Next, since the surface is compact,
there is a finite cover with triangles, and after refining where these
overlap we find that every compact connected 2-manifold is triangulable.
Finally, every connected triangulable 2-manifold is one of these.
Indeed, given a triangulation, we can order the triangles such that
each after the first has at least one edge in common with an earlier
triangle. By induction we see that the subspace on the union of an
initial part of this sequence of triangles is homeomorphic to a
plane regular polygon of which some pairs of edges are identified.
Thus, the entire manifold is isomorphic to a plane 2n-gon
of which the edges are identified in pairs. Going around the polygon,
we can write down a word in n variables x
where each occurs twice, either as x or as x^(-1),
written as x'.
For example, the word xyx'y' represents the torus.
Now we can juggle these words a bit, nothing that (i) cyclic
permutation of the word does not change the manifold,
(ii) abxcdx' = aydcy'b (where equality denotes
that the corresponding 2-manifolds are homeomorphic, and
a,b,c,d are arbitrary words),
(iii) abxcdx = aydb'yc'.
(Indeed, in both cases, let y be an edge from the point
where a meets b to the point where c
meets d, cut along y and glue along x.)
(iv) axx' = a (when a has length at least 4).
Suppose we let a end in a lot of pairs zz of two equal edges. When there are still more equal edges, we can apply (iii) to get the first element of a pair of two equals directly following a, and then once more (with empty b and d) to get both directly following a, and we can enlarge a. After doing this, start moving commutators [x,y] = xyx'y' to the tail of a. When there is an interlocking pair ...x...y...x'...y'..., use (ii) to cyclically rotate the parts between y and y' until x is in front of y and x' directly follows it, and then cyclically rotate the part between x' and x until y' directly follows x', and we have a commutator xyx'y' substring. After all this, our manifold descriptor consists of pairs xx and commutators xyx'y' and pairs xx'. Finally, if both a square and a commutator occur, we can turn this into 3 squares. Indeed, let us rename y to x again, and take two of b,c,d empty, then (iii) says abxx = axb'x = axxb. Now axx[y,z] = axz'y'xy'z' = axyx'yz'z' = axxyyz'z'.
Let Gp(x,y,...; e,f,...) denote the group defined by generators x,y,... and relations e = f = ... = 1, and let [x,y] denote the commutator xyx^(-1)y^(-1). Then
(i) P(M(g)) = Gp(x1,y1,...,xg,yg; [x1,y1], ..., [xg,yg])
(ii) P(N(h)) = Gp(x1,...,xh; (x1)^2, ..., (xg)^2).
A different way of distinguishing these surfaces is by use of the Euler characteristic and orientability. The Euler characteristic chi of a triangulable 2-manifold is the number of vertices minus the number of edges plus the number of triangles in any triangulation of the surface. This is well-defined: the number does not change upon refinement of the triangulation, and two triangulations have a common refinement.
We have chi(M(g)) = 2-2g and chi(N(h)) = 2-h. More generally, chi(M#N) = chi(M)+chi(N)-2.
A triangulable surface is called orientable if it has a triangulation such that we can orient the boundary of all triangles in a directed 3-cycle in such a way that if two triangles have an edge in common they induce opposite directions on this edge. Orientability is invariant under refinement, so orientability is also an invariant of the surface. A surface M#N is orientable if and only if both M and N are. Now the sphere M(0) and the torus M(1) are orientable, but the real projective plane N(1) is not.
In the proof we used compactness. Without it there are other examples. The simplest is the ordinary plane.
In the above we saw that nonhomeomorphic compact 2-manifolds are distinguished by their fundamental group. In particular, when a 2-manifold is homotopy equivalent to the sphere S(2), then it is homeomorphic to the sphere. It is unknown whether the same holds for 3-manifolds and S(3). This is the famous Poincaré Conjecture. (At first Poincaré conjectured that a compact 3-manifold with the same homology groups as a 3-sphere must be a 3-sphere, but he found a counterexample himself and changed the conjecture to say that any compact 3-manifold with vanishing fundamental group is the 3-sphere.) The 4-dimensional case was settled by Freedman (The topology of four-dimensional manifolds, J. Diff. Geometry 17 (1982) 357-453). All higher dimensional cases had been done already by Smale, see S. Smale, Generalized Poincaré's conjecture in dimensions greater than 4, Ann. Math. 74 (1961) 391-406, and A survey of some recent developments in differential topology, Bull. Am. Math. Soc. 69 (1963) 131-145. Both Smale and Freedman got a Fields medal.
Jules Henri Poincaré
Michael Hartley Freedman
An n-dimensional manifold with boundary is a Hausdorff space where each point has a neighbourhood isomorphic to an open set in a closed halfspace in Euclidean n-space E(n).
From the above we immediately obtain a classification of all 2-dimensional manifolds with boundary.
Indeed, the boundary will be a union of disjoint cycles, and if we glue a disk (a `cap') in the hole bounded by a cycle, we fill the hole. After filling all holes we obtain M(g) or N(h). Thus, the only additional invariant is the number of holes e.
Exercises (i) Show that a Möbius band (code xpxq)
is a real projective plane with one hole.
Show that M(g,e) (M(g) with e holes) has Euler characteristic 2-2g-e and that N(h,e) (N(h) with e holes) has Euler characteristic 2-h-e.