- TOPOLOGY^389
Β ΕDΒ'A'AFIGURE 15. Left: the projective plane triangulated and cut open.
If two opposite edges with corresponding endpoints are glued to-
gether, the figure becomes a Mobius band. In three-dimensional
space it is not possible to glue all the edges together as indicated.
Right: the Mobius band as originally described by Mobius.chains) C = n\C\ + · · · + nrCr of oriented curves Cj with integer coefficients Tij.
The interpretation of such a combination came from analysis: A line integral over
C was interpreted as the number I = n\I\ + • • · + nTIr, where Ij was the line
integral over Cj. When generalized to fc-dimensional manifolds (called varieties by
Poincare) and combined with the concept of the boundary of an oriented manifold
as a cycle, this idea was the foundation of homology theory: The fc-cycles (fc-chains
whose boundaries are the zero (fc — l)-chain—Poincare called them closed varieties)
form a group, of which the fc-cycles that are the boundary of a (fc + l)-cycle form
a subgroup. When two homologous cycles (cycles whose difference is a boundary)
are identified, the resulting classes of cycles form the fcth homology group. For ex-
ample, in the sphere shown in Fig. 14, the diagonal that is drawn forms a cycle.
This cycle is the complete boundary of the upper and lower triangles in the figure,
and it turns out that any cycle on the sphere is a boundary. The first homology
group of the sphere is therefore trivial (consists of only one element). For the torus
depicted in Fig. 14, a and b are each cycles, but neither is a boundary, nor is any
cycle ma + nb. On the other hand, the cycle formed by adding either diagonal to
a+b is the boundary of the two triangles with these edges. Thus, the first homology
group of the torus can be identified with the set of cycles ma + nb. Any other cycle
will be homologous to one of these.
Poincare also introduced the notion of the fundamental group of a manifold.
He had been led to algebraic topology partly by his work in differential equations.
He discovered the fundamental group by imagining functions satisfying a set of dif-
ferential equations and being permuted as a point moved around a closed loop. He
was thus led to consider formal sums of loops starting and ending at a given point,
two loops being equivalent if tracing them successively left the functions invari-
ant. The resulting set of permutations was what he called the fundamental group.
He cautioned that, despite appearances, the fundamental group was not the same
thing as the first homology group, since there was no base point involved in the