Invariants of Fold-maps via Stable Homotopy Groups

Abstract

In the 2-jet space J 2 (n, p) of smooth map germs (R n , 0) → (R p , 0) with n ≥ p ≥ 2, we consider the subspace Ω n−p+1,0 (n, p) consisting of all 2-jets of regular germs and map germs with fold singularities. In this paper we determine the homotopy type of the space Ω n−p+1,0 (n, p). Let N and P be smooth (C ∞ ) manifolds of dimensions n and p. A smooth map f : N → P is called a fold-map if f has only fold singularities. We will prove that this homotopy type is very useful in finding invariants of fold-maps. For instance, by applying the homotopy principle for fold-maps in [An3] and [An4] we prove that if n − p + 1 is odd and P is connected, then there exists a surjection of the set of cobordism classes of fold-maps into P to the stable homotopy groupHere, ν k P is the normal bundle of P in R p+k and γ G n−p+1, denote the canonical vector bundles of dimension over the grassman manifold G n−p+1, . We also prove the oriented version.

Cobordisms of maps with singularities of given class

“…Although available only in the case of fold-maps, in Section 10 we will construct a simpler spectrum associated to special generic maps, say definite fold-maps, whose stable homotopy group is a direct summand of C.n; P I i;0 ; i;0 / and also construct the corresponding classifying space. This result is a refinement of [5,Theorem 0.3]. We should note that special generic maps do not satisfy the h-principle (see Burlet and de Rham [14] and Saeki and Sakuma [49]).…”

“…These theorem show the importance of the homotopy type of Ω I (n, p). In [6], [7], [8] and [9] we have determined the homotopy type of Ω (i,0) (n, p) for i = max{n − p + 1, 1}, and studied NCob (Ω (i,0) n ,Ω (i,0) n+1 ) n,P and OCob…”

“…Here, we prove the following refinement of [5,Theorem 0.3]. This theorem should be compared with the results of Kalmár [27; 28].…”

“…Since P 1 (τ ( V , T (f )) = 0, it follows from [34, Proof of Lemma 2] that T V is trivial on CV and the 4-skeleton of V . Then the assertion follows from π i (SO(8)) = 0 for i = 4, 5, 6 by applying the obstruction theory for p SP n : M(n + 2) → SP n with fiber SO (8).…”

A homotopy principle for maps with prescribed Thom-Boardman singularities

“…Theorem 0.1 is very important, even for fold-maps, in proving the relations between fold-maps, surgery theory and stable homotopy groups ([An4, Theorem 1], [An5,Theorems 0.2 and 0.3] and [An8]), where the homotopy type of Ω n−p+1,0 (n, p) determined in [An3] and [An5] has played an important role. We can now readily deduce the famous theorem about the elimination of cusps in [L1] and [E1] (see also [T]) from these theorems.…”