The most important object in real singularity theory is the C∞ map germ and the
most important equivalence relation among them is C∞ right left equivalence. In [7],
we presented a new systematic method for the classification of C∞ map germs by
characterising C∞ right left equivalence. This paper is a topological version of [7].
Two C∞ map germs
f, g:(Rn, 0) →
(Rp, 0) are said to be topologically equivalent
if there exist homeomorphism map germs
s:(Rn, 0) → (Rn 0) and
t:(Rp, 0) → (Rp, 0)
such that f(x) = t∘g∘s(x).
The notion of topological equivalence, although it
seems to be unnatural, is also important since we know the existence of C∞ moduli
for the classification of C∞ map germs with respect to C∞ right left equivalence.
However, we had only one method to obtain topological equivalence for two given
C∞ map germs, as stated in the following.
For two given C∞ map germs
f, g:(Rn, 0) →
(Rp, 0), take an appropriate one-parameter family
F:(Rn×[0, 1],
{0}×[0, 1]) → (Rp, 0)
such that F(x, 0) = f(x) and
F(x, 1) = g(x). Then prove that F is in fact
topologically trivial.(*)
Two C∞ map germs
f, g:(Rn, 0) →
(Rp, 0) are said to be [Kscr ]-equivalent if there
exist a C∞ diffeomorphism map germ
s:(Rn, 0) →
(Rn, 0) and a C∞ map germ
M:(Rn, 0) → (GL(p, R), M(0))
such that f(x) = M(x)g(s(x)).
The notion of [Kscr ]-equivalence was introduced by Mather [4, 5]
in order to classify the C∞ stable map
germs, and we know that generally in a [Kscr ]-orbit there are uncountably many C∞ right
left orbits.
Hence it is significant to give an alternative systematic method for the topological
classification even in a single [Kscr ]-orbit, which is the purpose of this paper. One of our
results (Theorem 1.2) yields the following well-known theorem [2] as a trivial
corollary.
