Colloquium publications.

ANALYSIS SITUS.

113

symbols for the boundaries of the oriented n-cells of rn and
whose last column is the sum of the columns of En. Since each
column of En contains either two + l's, two - l's or one + 1
and one - 1, the elements of the last column of En Dn are either
0 or =t 2. They cannot all be 0 because the rank of En is an.
Hence
En, Dn = Cn-l1-En*
where Cn,_1- is a square matrix of determinant unity and En* is
a matrix all of whose elements are zero except those of the main
diagonal. The elements of the main diagonal are all ~t 1 except
the last which is 4 2.
Relation between the Betti Numbers and the Connectivities
23. The matrices Ek reduce to the matrices Hk if all elements
are reduced modulo 2. Hence if 3k denote the number of even
k-dimensional coefficients of torsion, the ranks of Ek and Hk are
connected by the relation
rk - Pk = ck-1.
Since
Rk - 1 = COk - Pk - Pk+l
and
Pk - 1 = ak - rk - rk+1
it follows that
(15) Rk- Pk = 8k-1 + 5k
which is the formula for the connectivities in terms of the Betti
numbers and the coefficients of torsion.
24. In the Monatshefte fur Math. und Physik, Vol. 19 (1908),
p. 49, a set of numbers, Qk (k = 0, 1, 2,..., n), are defined by H.
Tietze in terms which are very similar to our definition of the
numbers Rk. But Tietze finds the formula (p. 56):
Qk = Pk + 8k-1
which shows that the Qk's as he used them are distinct from the
Rk'S.