Colloquium publications.

ANALYSIS SITUS.

103:

This completes the proof that the two theorems (1) and (2)
of ~ 1 are true for Cn if they are true for all Ci (i < n) and thus
establishes the cycle of theorems and definitions in ~ 1 for all
values of n.
Matrices of Orientation
5. Each column of the matrix Hk (k = 1, 2, ***, n) for Cn is
the symbol in the sense of ~ 2 for a (k - 1)-circuit bounding a
k-cell. This (k - 1)-circuit is orientable because the set of
points on it is a (k - 1)-dimensional sphere. Hence by changing
some of the l's in the symbol (xl, x2, * ', xa) for the k-circuit
into - l's this symbol is converted into the symbol in the sense
of ~ 2 for one of the two oriented (k - 1)-circuits which can be
formed from the k-dimensional circuit. Hence by changing some
of the l's of Hk into - l's Hk can be converted into a matrix,
Ek = | I ijkl | (i = 1, 2, ', Oak; j = 1, 2, '* ', Oak-1)
the jth column of which represents the (k - 1)-circuit which is
associated with ajk to form the oriented i-cell crjk.
The oriented (k - 1)-cells which enter into this (k- 1)circuit are said to be positively related to ojk and negatively related
to - ajk, while their negatives are said to be negatively related
to ajk and positively related to - ajk. Hence the matrix Ek is
such that E jk = 1 if aik-1 is positively related to o-j, eijk =- 1
if aik-l is negatively related to a-k and Eik = 0 if aik-l is neither
positively nor negatively related to o-k. If the notation is changed
so as to interchange the meanings of ajk and - ajk, the elements of the jth column of Ek and of the jth row of Ek+1 must
be multiplied by - 1.
6. In case Cn is an n-circuit the rank of En determines whether
Cn is orientable or not. For if Cn is orientable each orientable
(n - 1)-cell is positively and negatively related to equal numbers
of orientable n-cells and hence each column of En contains equal
numbers of + l's and - l's. Hence the rank of En is at most
an - 1. It cannot be less than an - 1 because then the rank
of Hn would be less than an - 1 (Cf. ~ 55, Chap. II).
On the other hand if the rank of En is an - 1 there must be a