Colloquium publications.

ANALYSIS SITUS.

135

20. The operation of shrinking a 1-cell ai1 of a linear graph
Ci to a point changes Ci into a complex Cl' having the same
group as C1. This is because: (1) if a closed curve is entirely
on ail it can be deformed into coincidence with a single point
and (2) if a closed curve C on C1 cannot be deformed into coincidence with a point, the operation of shrinking ail to a point
converts C into a curve which cannot be deformed to a point
on Ci'. From this it follows that: (1) The group of any tree
is the identity; (2) the group of any complex which is not a
tree is the same as the group of a complex consisting of R1 - 1
1-cells each having a 0-cell 0 as its initial and terminal point,
and no two having a point in common. This group consists of
R - 1 operations gi (i = 1, 2, **, R- 1) and all combinations of them. Thus the general expression for an operation of
the group is
glal g2a2.. ga'.g g 2 * * gM.. gll * g22. * gj1a
where the exponents can be any integers, positive, negative or
zero, and = R, - 1,
21. The operations gi, g2,, g, are called the generators of
the group of C1. They are absolutely independent of each
other, that is to say they satisfy no identities except the laws of
combination given in ~ 16.
In the general theory of discrete groups having a finite number
of generators the generators are supposed to satisfy certain
identities of the form,
giml gi2M... gi m- 1
which are known as generating relations. The groups of n-dimensional complexes (n > 2) will be seen usually to have generating
relations. The group of a linear graph is thus characterized
by the lack of generating relations.
The Group of a Two-dimensional Complex
22. Let C2 be a two-dimensional complex, C2 a regular subdivision of it, and let Cl be the linear graph composed of the