Colloquium publications.

92

THE CAMBRIDGE COLLOQUIUM.

give
ao - al + a2 -. * * - an = 0,
as already noted in ~ 9.
Generalized Manifolds
31. It follows from ~ 24 that the cells of higher dimensionality
incident with any cell of a complex defining a manifold constitute
a simply connected star. This property could in fact be taken
as the definition of a manifold. It is the basis of the following
generalization.
A generalized manifold of n dimensions is the set of all points on
an n-circuit C, such that if ai-1 is any cell of Cn the incidence
relations among the (i)-cells, (i + 1)-cells,.*, (i + k)-cells
(where i + k = n) incident with ai-1 are the same as the incidence
relations among the 0-cells, 1-cells,.*, k-cells of a complex
defining a generalized manifold of k dimensions; a generalized
manifold of zero dimensions is a 0-circuit.
This definition is obviously invariant under the group of all
homeomorphisms. For n = 0, 1, 2, a generalized manifold is
the same as a manifold. But for n > 3 it includes sets of points
which are not manifolds in the narrow sense.
32. To bring this out let us consider the following example
given in the article on Analysis Situs by Dehn and Heegaard in
the Encyclopadie. Let S4 be a Euclidean space of four dimensions, a~ a point in S4, S3 a three-space in S4 but not on a~, and
M2 an arbitrary two-dimensional manifold (e.g., an anchor ring)
in S3. Let M2 be decomposed into 0-cells, 1-cells and 2-cells
constituting a two-dimensional complex, B2. The segment
joining any 0-cell of B2 to a~ is a 1-cell, the points on the segments joining the points of a 1-cell of B2 to a~ constitute a 2-cell,
and the points on the segment joining the points of a 2-cell of
B2 to a~ constitute a 3-cell. The complex C3 composed of all
the' 1-cells, 2-cells and 3-cells found by this process, together
with a~ and the cells of B2, is such that the boundary of an arbitrarily small neighborhood of a~ is of the same structure as B2.
Hence the set of points on each such boundary is a surface like
M2 (e.g., an anchor ring).