Colloquium publications.

70

THE CAMBRIDGE COLLOQUIUM.

defining these manifolds are orientable. In like manner, the
manifolds defined in ~ 26 are non-orientable.
Normal Forms for Manifolds
61. It has now been proved that any two homeomorphic
manifolds are both orientable or both one-sided, and have the
same connectivity. Conversely it can be proved that if two
closed manifolds are both orientable (or both one-sided) and have
the same connectivity they are homeomorphic. In other words, R1
and the orientableness of a closed manifold characterize it completely from the point of view of Analysis Situs.
62. By way of establishing this theorem we shall outline a
method of reducing any manifold to a normal form. This
reduction is related to that given in ~ 31, Chap. I for the matrix
H1. It was there found that there are two matrices Ao and B,
such that
Ao-1.H * B1= Hi*
where Hi* is a matrix of unitary type.
In the case of a single manifold, which we are now considering,
Ro = 1 and
A = ai - pi = R1 - 1 + P2.
The matrix B1 is such that its first pi columns are the symbols for
the 1-cells of a tree, T1, and its last ai - pi columns are the symbols for a complete set of 1-circuits C11,, 2..., C1'. The
1-cells not in the tree T1 were denoted (~ 26, Chap. 1) by ap'
(p = jl,, j2'', J) and are such that the circuits C1i, C12, *.,
Car could be formed by adjoining ajl1, aj1,.* * successively to T1.
By reference to ~ 27, Chap. 1 it is clear that C1 may be taken
to be a bounding circuit and aj1 to be a 1-cell of Ci1. Similarly,
if P2 > 2, C12 may be taken as bounding; and by repeating the
argument it is found that ap1 (p = jl, j2,.* *, jp,) may be chosen
so that C1, C12,.., CP2 are all bounding circuits. The remaining circuits C1P2+1,..., C1 are necessarily non-bounding since
the number of bounding circuits in a complete set is p2.
63. The tree T1 and the R1 - 1 1-cells ap1 (p = jp2+i 'I, j)