Colloquium publications.

88

THE CAMBRIDGE' COLLOQUIUM.

termined in ~ 16. Hence the value of R, determined by Cn is
not greater than that determined by Cn. It also cannot be less,
for if so there would be a linear relation among the i-circuits
Ci (p = 1, 2,..., Ri - 1) regarded as circuits of Cn. But this
would mean that there was a complex Kj+j composed of cells of
Cn and bounded by some or all of the circuits Ci7. By an argument like that in the paragraph above Kj+j could be replaced
by a complex K',+ which contains none of the vertices Pjn of
Cn and which therefore is composed of cells of Cn. But the existence of Ki+2 would mean a linear relation among the i-circuits
of CiP regarded as z-circuits of Cn. Hence the value of Ri determined by Cn is not less than that determined by Cn.
Manifolds
24. By a neighborhood of any i-cell ai on a complex Cn is meant
any set S of non-singular cells on Cn such that any set of points
of Cn having a limit point on ai contains points on the cells of S.
If Cn is an n-circuit such that every point on it has a neighborhood which is an n-cell the set of points on Cn is called a closed
n-dimensional manifold. A set of points obtainable by removing
a finite number of n-cells and their boundaries from a closed
manifold is called an open manifold of n dimensions.
If P is an arbitrary point of a complex Cn it can be taken as
one of the points P/ in a regular subdivision of Cn. The set of
cells of Cn which are incident with P determines a neighborhood
of P. Hence the process of regular sub-division gives an explicit
method of determining whether the set of points on Cn is or is
not a manifold.
Dual Complexes
25. A complex Cn' is said to be dual to a complex Cn if the
incidence relations between the k-cells and (k - 1)-cells of Cn'
are the same as those between the (n - k)-cells and (n - k + 1)cells of Cn for k = 1, 2,...*, n. In case Cn defines a manifold,
a complex Cn' dual to Cn can be constructed by first making a
regular subdivision of Cn into Cn, then defining as an n-cell of CY'
the set of all points on each star of cells of Cn having a vertex of