Colloquium publications.

48

THE CAMBRIDGE COLLOQUIUM.

25. Let a sphere, S, be decomposed into cells by the process
described in ~ 11 and let s12, s22, * *, Sp2 be p of the 2-cells so
obtained. Let T1, T2, *., TP be p anchor rings no two of which
have a point in common and which are such that si2 (i = 1, 2,
* *, p) is a 2-cell of Ti while Ti and S have no other points in
common than those of si2 and its boundary. The set of all points
on the 2-circuit,
(1) M2 = S + T1+ T2+.. + Tp (mod. 2),
is called a sphere with p handles, or an orientable manifold of genus
p, or an orientable manifold of connectivity 2p + 1. The proof
that the set of points on M2 is a manifold is made by subdividing
it into 2-cells. Bythe same device it is easy to prove that a
sphere with one handle is an anchor ring.
26. If one of the anchor rings Ti in the last section is replaced
by a projective plane, the 2-circuit 11~2 is easily seen to define a
manifold. We shall refer to this as a one-sided manifold of the
first kind of genus p - 1, or of connectivity 2p. It is easy to
verify that a projective plane is a one-sided manifold of the first
kind of genus zero.
If two of the manifolds Ti are projective planes and the rest
are anchor rings the 2-circuit M2 again defines a manifold. This
is called a one-sided manifold of the second kind of genus p - 2,
or of connectivity 2p - 1.
In this section and the last one the terms connectivity and
genus are used in such a way that
R1- 1 = 2p + k
where R1 is the connectivity, p is the genus, and k = 0 for an
orientable manifold, k = 1 for a one-sided manifold of the first
kind, and k = 2 for a one-sided manifold of the second kind.
27. The fundamental problem of two-dimensional Analysis
Situs is that of classifying all two-dimensional manifolds. The
solution of this problem is found by proving: (1) that for every
manifold there is an integer R1, the connectivity (cf. ~ 29), which
is an invariant under the group of all homeomorphisms; (2) that