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.

**Proof**
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

**Theorem**
(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.

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*.

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