Colloquium publications.

124

THE CAMBRIDGE COLLOQUIUM.

we have
EiEi+i= 0
and therefore the matrices E1, E2, E,, E, satisfy the conditions
required of the matrices of orientation of Cu'.
Since the invariant factors of Ek+1' are the same as those of
Ek+l and En-k = Ek+l' it follows that the invariant factors of
En-k and of Ek+l are the same. Hence the (n - k - )-dimensional coefficients of torsion of an oriented manifold are the same
as the k-dimensional ones. In other words
tjk = tn-k-1 (k = 1, 2,.., n - 1; = 1, 2,, ).
40. In view of the equation (~ 23),
Rk - Pk = 8k-1 +6kc
and the equation Rn-k = Rk it follows from this that the Betti
numbers of an orientable manifold satisfy the condition
Pn-k = Pk.
It should be noted particularly that while the relation Rn-k
= Rk is satisfied by the connectivities of any manifold the relation Pn-k = Pk is restricted to the orientable manifolds.