Colloquium publications.

148

THE CAMBRIDGE COLLOQUIUM.

form by a process analogous to that outlined in ~ 62, Chap. II.
If we start with a non-singular complex C3 defining an orientable
manifold M3 and perform a sequence of operations (1) of coalescing pairs of 3-cell which have a common 2-cell on their boundaries
and (2) of shrinking to a point 1-cells which join distinct points,
C3 is reduced to a complex C3' consisting of one 3-cell and one
0-cell and equal numbers, a, of 2-cells and 1-cells. Hence M3
may be represented by means of the interior and boundary of a
Euclidean sphere, the boundary being a map all the vertices of
which represent the same point of M3. The 2-cells of this map
fall into a pairs each of which represents a single 2-cell of C3'.
The 1-cells of the map fall into a sets such that all 1-cells in the
same set represent the same 1-cell of C3'.
41. This representation of a manifold M3 by means of a sphere
has not yet proved as fruitful as the related Heegaard diagram
which may be obtained as follows: Let the 0-cell of C3' be
enclosed by a small 3-cell containing it and let each of the 1-cells
of C3' be enclosed by a small tube containing it. Thus we
obtain a three-dimensional open manifold L3 bounded by a twodimensional manifold M2 consisting of a sphere with a handles.
M3 is orientable if and only if M2 is orientable. In case M3 is
orientable L3 can be represented as the interior of a sphere with
handles having no knots or links in a Euclidean 3-space.
The 2-cells of C3' meet M2 in a system of a curves no two of
which intersect and which bound a set of a 2-cells a12, a22, * *, aa2
contained in the 2-cells of C3'. The points of the 2-cells a 2
together with the points of the 3-cell of C3' which are not in L3
or M2 constitute the interior of an open three-dimensional manifold N3 bounded by M2.
Thus M3 consists of two open manifolds L3 and N3 which
have a common boundary, M2. It is clear that M3 is fully
determined if L3, M2 and the boundaries c1, c2,., Ca of the
cells ai2 are given. For the manifold M3 can be reconstructed
by putting in 2-cells bounded by the curves c1, c2,., Ca and a
3-cell bounded by M2 and these 2-cells, each counted twice.
The representation of a manifold by means of L3, M2, and
cl, c2,.., ca is called the Heegaard diagram. It is due (in a