- TOPOLOGY 387
in which æ — w^2 — 0, every complex number æ = a + bi has two distinct complex
square roots:w = ±(u + IV), where u = W and í = sgn (b)]l.The square roots of the positive real numbers that occur here are assumed positive.
There is no way of choosing just one of the two values at each point that will result
in a continuous function w — \J~z. In particular, it is easy to show that any such
choice must have a discontinuity at some point of the circle \z\ — 1.
One way to handle this multivaluedness was to take two copies of the z-plane,
labeled with subscripts as z\ and z-i and place one of the square roots in one plane
and the other in the other. This technique was used by Cauchy and had been
developed into a useful way of looking at complex functions by Victor Puiseux
(1820-1883) in 1850. Indeed, Puiseux seems to have had the essential insights that
can be found in Riemann's work, although differently expressed. Riemann is known
to have seen the work of Puiseux, although he did not cite it in his own work. He
generally preferred to work out his own way of doing things and tended to ignore
earlier work by other people. In any case, the essential problem with choosing one
square root and sticking to it is that a single choice cannot be continuous on a
closed path that encloses the origin without going through it. At some point on
such a path, there will be nearby points at which the function assumes two values
that are close to being negatives of each other.
Riemann had the idea of cutting the two copies of the z-plane along a line
running from zero to infinity (both being places where there is only one square
root, assuming a bit about complex infinity). Then if the lower edge of each plane
is imagined as being glued to the upper edge of the other,^43 the result is a single
connected surface in which the origin belongs to both planes. On this new surface
a continuous square-root function can be defined. It was the gluing that was really
new here. Cauchy and Puiseux both had the idea of cutting the plane to keep a
path from winding around a branch point and of using different copies of the plane
to map different branches of the function.
Riemann introduced the idea of a simply connected surface, one that is dis-
connected by any cut from one boundary point to another that passes through its
interior without intersecting itself. He stated as a theorem that the result of such a
cut would be two simply connected surfaces. In general, when a connected surface
is cut by a succession of such crosscuts, as he called them, the difference between
the number of crosscuts and the number of connected components that they pro-
duce is a constant, called the order of connectivity of the surface. A sphere, for
example, can be thought of as a square with adjacent edges glued together, as in
Fig. 14. It is simply connected because a diagonal cut disconnects it. The torus,
on the other hand, can be thought of as a square with opposite edges identified
(see Fig. 14). To disconnect this surface, it is necessary to cut it at least twice, for
example, either by drawing both diagonals or by cutting it through its midpoint
with two lines parallel to the sides. No single cut will do. The torus is thus doubly
connected.
(^43) You can visualize this operation being performed if you imagine one copy of the plane picked
up and turned upside down above the other so that the upper edge is glued to the upper edge
and the lower to the lower.