The purpose of this note is to generalise, and give a more illuminating
proof, of
a theorem of [13] (Theorem 1.1 below). Before stating
it, we provide some
introductory information. Consider the following two sequences of pictures:
in each
we see a 1-parameter family Xℝ,t
of real algebraic hypersurfaces, which undergoes a
bifurcation when the parameter t is equal to 0. Note that in Figure
1, both (i) (a) and
(i) (b), and in (ii) (b), the surface Xℝ,t
has a purely 1-dimensional part, which we have
indicated with a dotted line, and that in (i) (b) we have drawn a curve
vertically along
the middle of the surface to make clearer the way it passes through itself.
The reader
will observe that in (a) the surface Xℝ,t
is homotopically a 2-sphere when t>0 and
a 0-sphere when t<0, while in (b) Xℝ,t
is a homotopy 1-sphere both for t<0 and
t>0.
Such sequences are typical in singularity theory; each is in fact the
family of
algebraic closures of images of a versal deformation of a codimension 1
singularity
of mapping.
Now suppose that the complexification X[Copf ],t
is a homotopy n-sphere. In [13] the
second author pointed out that it follows that Xℝ,t
is a homotopy sphere for t≠0
(allowing the empty set as a −1-sphere). Indeed, in the local situation,
or globally in
the weighted homogeneous case, there are well-defined integers
k+ and k− between −1
and n such that
Xℝ,t≃Sk+
for t>0
and Xℝ,t≃Sk−
for t<0.
We describe Xℝ,t for
t∈ℝ−0 as ‘good’ if the homotopy dimension
of Xℝ,t is equal to n. In this
case the inclusion
Xℝ,t[rarrhk ]Xt
is a homotopy equivalence [13, 1.1].
