such that ( the image of fr) =
for r = 1,2,3, and such that
f ° fr = fr°q
where q : Cr - T(r) is the covering map. The image of fr is the r-decker set.
It is convenient (and we do so from now on) to include the branch points among
the double points. In this way the boundary of the double point manifold consists
of branch points. The triple points are in the image of /2, and so each triple point
has three pre-images in the double point manifold. Observe that if y £ R 3 is a
branch point, then f~1(y) contains one point. We will often abuse the notation
and call either y or f~l(y) a branch point. We include f~x(y) in the double decker
manifold, and extend the covering q to a branched covering.
Broken surface diagrams. Let a generic surface / : F R
be given.
Consider y G R
to be a singular point of f(F); the point y is either a double
point, triple point, or branch point of the mapping. Let N(y) denote a 3-ball
neighborhood of y. A local broken surface diagram near y is formed by replacing
the intersection f(F) D N(y) with a surface with boundary of one of the following
forms: If y is a double point, then we replace f(F) D N(y) with three embedded
disks by removing an open tubular neighborhood of the double decker arc on one
of the sheets forming the double point set. If y is a triple point, then we replace
the intersection with seven embedded disks, and if y is a branch point we replace
the intersection with a single embedded disk with y on the boundary. Local broken
surface diagrams are depicted in Fig. 2.
A broken surface diagram consists of a generic surface in R
where each singular
point has a neighborhood which has a local broken surface diagram.
Some explanations are in order. Figure 2 expresses the idea that an open
regular neighborhood of an arc of the double decker curve is removed from one of
the intersecting sheets near the singular point. If the singular point is a double
point, this is precisely what happens. If the singular point is a triple point, then
one of the three sheets has a neighborhood of its coordinate axes removed (under
the identification with the standard coordinate ball), another sheet has only the
neighborhood of one axis removed, and the third sheet remains intact. If y is
a branch point, then the neighborhood that is removed is shaped like a triangle
with y at a vertex. Thus a broken surface diagram is obtained by patching these
neighborhoods compatibly. Necessarily, the choices of broken sheets match along
the double arcs. In other words, if y,y' are double points and N(y) D N(yf) ^ 0,
let y" be a double point in this intersection with neighborhood N(y"). There is
double arc that passes through y, y", and y'. The sheet of f(F) that has its regular
neighborhood removed is the same sheet at all three double points.
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