Colloquium publications.

140

THE CAMBRIDGE COLLOQUIUM.

obtained from any other point 0'), the group G has no reference
to any particular point 0. This is because the terms lj and 11
in (3) cancel out when the assumption of commutativity is
introduced.
27. The fundamental group G is such that
gx= 1
signifies that the closed curve represented by gx bounds a 2-cell
on C2. The geometric significance of the group G is equally
simple. If gy is an element of this group
(6) gy = 1
signifies that the closed curve or set of closed curves represented
by gy bounds a two-dimensional complex on C, or in other words,
(7) gy - 0
where we now let gy stand for the oriented curve obtained by
identifying the initial and terminal points of the oriented 1-cell gy.
That (6) and (7) have the same geometrical significance is
immediately evident if one compares the steps by which (6) is
obtained from (5) with those by which (7) is obtained from the
fundamental homologies of ~ 28, Chap. IV.
Equivalences and Homologies
28. The operation of combining two elements of a group which
is called multiplication in the sections above can equally well
be denoted by the sign + and called addition. This is done in
fact by Poincare in a number of places. He thereby replaces
any relation of the type (1) by
(8) aill + ai2g2 + * * * + aingn +* * * + jigl + jig2
* * + jingn 0
which he calls an equivalence. In an equivalence the operation
of addition is non-commutative. The equivalence (8) signifies
that the elements on the left-hand member constitute the boundary of a 2-cell. To develop the theory of equivalence further