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.… Show more
“…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]).…”
Section: Introductionmentioning
confidence: 94%
“…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…”
Section: Let Imentioning
confidence: 99%
“…Here, we prove the following refinement of [5,Theorem 0.3]. This theorem should be compared with the results of Kalmár [27; 28].…”
Section: Nc1 G Nc1mentioning
confidence: 99%
“…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).…”
Let N and P be smooth closed manifolds of dimensions n and p respectively. Given a Thom-Boardman symbol I, a smooth map f : N → P is called an Ω I -regular map if and only if the Thom-Boardman symbol of each singular point of f is not greater than I in the lexicographic order. We will represent the group of all cobordism classes of Ω I -regular maps of n-dimensional closed manifolds into P in terms of certain stable homotopy groups. As an application we will study the relationship among the stable homotopy groups of spheres, the above cobordism group and higher singularities.
“…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]).…”
Section: Introductionmentioning
confidence: 94%
“…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…”
Section: Let Imentioning
confidence: 99%
“…Here, we prove the following refinement of [5,Theorem 0.3]. This theorem should be compared with the results of Kalmár [27; 28].…”
Section: Nc1 G Nc1mentioning
confidence: 99%
“…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).…”
Let N and P be smooth closed manifolds of dimensions n and p respectively. Given a Thom-Boardman symbol I, a smooth map f : N → P is called an Ω I -regular map if and only if the Thom-Boardman symbol of each singular point of f is not greater than I in the lexicographic order. We will represent the group of all cobordism classes of Ω I -regular maps of n-dimensional closed manifolds into P in terms of certain stable homotopy groups. As an application we will study the relationship among the stable homotopy groups of spheres, the above cobordism group and higher 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.…”
Abstract. Let N and P be smooth manifolds of dimensions n and p (n ≥ p ≥ 2) respectively. Let Ω I (N, P ) denote an open subspace of J ∞ (N, P ) which consists of all Boardman submanifolds Σ J (N, P ) of symbols J with J ≤
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