Colloquium publications.

ANALYSIS SITUS.

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which are obtained from (4) of ~ 30 by the transformation
(13).
38. It has thus been proved that the coefficients of torsion
tlk, t2k,.*, trk are uniquely determined and are the same for all
sets of congruences
A/ — 0
defined as in ~ 35. Since Pk - 1 = U - rk it follows that the
Betti number Pk is also the same for all these sets of congruences.
But since congruences and homologies are obviously transformed
into congruences and homologies by any homeomorphism, it
follows that the Betti numbers and the coefficients of torsion are
Analysis Situs invariants.
There is no difficulty in seeing that the Betti numbers and
coefficients of torsion of Cn and of Cn are the same. Hence
these numbers are the same for all complexes into which C, can
be decomposed.
Duality of the Coefficients of Torsion
39. The duality relation,
Rn-i = Ri (i = 0, 1,, n- 1)
was proved (~ 29, Chap. III) by showing that if Hk (k = 1,.., n) are the incidence matrices of Cn and Hk (i = 1, 2,. *, n)
those of a matrix Cn' dual to Cn, then
Hn-i = Hi+I' (i = 0, 1,,n - i),
where Hi+1' is the matrix obtained by interchanging the rows and
columns of Hi+~.
Now if Cn is an orientable manifold the matrices Ek (k = 1, 2,.., n) can be formed from Hk in such a way that the matrix H,
has one + 1 and one - 1 in each row. Moreover the equation
Ek'Ek+1 = 0
is equivalent to
Ek+1'*Ek' = 0.
Hence if we introduce - signs in the matrix H k to define Ek so that
En-i = Ei+l'