I. EMBEDDING FINITE COVERS INTO BUNDIES
1. RENDVTNG SlhGULARTITES OP MAPS
All manifolds considered in this paper will be either smooth or PL (=piecewise
linear). Probably most of the results are true for topological manifolds  but in-
vestigations along these lines would carry the authors far afield from the original pro-
ject (for example,we would have to consider the existence of cross-sections of Euclidean
bundles which are contained in normal microbundles ). Recall that a smooth structure
on a manifold induces a unique PL structure ; whenever we enter the realm of PL
topology with a smooth manifold we assume that it has this PL structure.
We shall appeal often to Hudson1s book for results in PL topology; and
are good alternative sources.
The authors are unable to cite a single reference for orientations, local orienta-
tions and intersection numbers which contains all the results which are needed for this
paper. and  contain many results which are needed.
To avoid a plethora of notation, we shall often leave out the basepoint in denoting
the fundamental group. In most cases,the choice of basepoints is irrmaterial except that
they be preserved under mappings for which we look at the induced homomorphisms of the
Quite often we shall perform the following construction. Let a, 3: [0, 1] -•X be
two paths such that x(l)= 3(0) and a(0)= 3(1) ; then a U 3 will denote the loop
(a map of [0, 1] into X which takes 0 and 1 into the same point) defined by
a U 3(t) = a(2t)
= 3(2t - 1)
Let a: [0, 1]+ X be the path defined by a(t) = o(l - t) . If f:X-^Y, then we
usually denote the composition f o (*U 3) by f (a U 3) . it will also be convenient to
let a denote the image of a .
We adopt the following terminology of Whitney . A mapping f: M"-*B n of a
compact n-dimensional manifold into Euclidean 2n-space is completely regular if (1) f
J±t l 1