Colloquium publications.

ANALYSIS SITUS.

91

Hence aik is incident with ajk+1 if and only if bi-k is incident with
bjn-k-1.
Duality of the Connectivities R,
29. Stating this result for the case k = n - 1, we have that
ai1 i inc nt w is incident with if and only if b is incident with b.
Hence the matrix of incidence relations between the O-cells and
1-cells of the complex Cn' is the matrix Hn' obtained from the
matrix Hn of the complex Cn by interchanging rows and columns.
In like manner it is seen that, in general, the matrix of incidence
relations between the (n - k - 1)-cells and (n - k)-cells of the
complex Cn' is the transposed matrix Hk' of the matrix Hk of
the complex Cn. Hence the matrices of incidence Hi, H2,..,
Hn of Cn' are the matrices Hn', Hn-1',. *, Hi' of Cn.
The ranks of these matrices are Pn, pn-, ', pi respectively.
Moreover the numbers of O-cells, 1-cells,. -- n-cells of Cn' are
an, an-l, * *, ai, ao respectively. Hence by the formula for the
i-dimensional connectivity Ri, it follows that the 1-, ***,
(n - 1)-dimensional connectivities of Cn' are Rn-1, ''*, R
respectively.
It was shown in ~ 23 that the connectivity Ri of a complex Cn
obtained by a regular subdivision of Cn is the same as that of Cn.
But by comparing ~ 22 with ~ 26 it is seen that Cn is a regular
subdivision both of Cn and of Cn'. Hence the connectivity Ri of
Cn' is the same as that of Cn. Hence Rn-1, Rn-2, ''', R1 are
the same as R1, R2,.., Rn-1, respectively. That is
Rn-k = Rk (k = 1, 2,.. - 1).
It should be noted that this duality relation does not apply to
Ro and Rn. In the case of a manifold, which we are considering
here, Ro = 1 and Rn = 2.
30. An important corollary of this result is that for a manifold
of an odd number of dimensions the characteristic is zero. For
the equations
n —1
ao - al +. ( + (- 1)n = 1+ (-l + (-1) i(Ri —1)
i-=1
and
Ri = Rn- (i= 1,2, *, n- 1)