There is a homotopy equivalence ϕ:M→M′ between closed smooth manifolds of an odd dimension such that ϕ*TM′, TM are stably isomorphic but not isomorphic to each other.

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.