We obtain complete geometric invariants of cobordism classes of oriented simple fold maps of (n+1)-dimensional manifolds into an n-dimensional manifold N in terms of immersions with prescribed normal bundles. We compute that this cobordism group of simple fold maps is isomorphic to the direct sum of the (n-1)th stable homotopy group of spheres and the (n-1)th stable homotopy group of the infinite dimensional projective space. By using geometric invariants defined in the author's earlier works, we also describe the natural map of the simple fold cobordism group to the fold cobordism group by natural homomorphisms between cobordism groups of immersions. We also compute the ranks of the oriented (right-left) bordism groups of simple fold maps.
Fold maps of (n + 1)-dimensional manifolds into n-dimensional manifolds are smooth singular maps which have the formula f (x 1 , . . . , x n+1 ) = (x 1 , . . . , x n-1 , x 2 n ± x 2 n+1 ) as a local normal form around each singular point. Fold maps can be considered as the natural generalizations of Morse functions. Let f : Q n+1 → N n be a fold map. The set of singular points of the fold map f is a two codimensional smooth submanifold in the source manifold Q n+1 and the fold map f restricted to its singular points is a one codimensional immersion into the target manifold N n . This immersion together with more detailed informations about the neighbourhood of the set of singular points in the source manifold Q n+1 can be used as a geometric invariant (see [13] and Section 1.3 of the present paper) of fold cobordism classes (see Definition 1.1) of fold maps, and by this way we obtain a geometric relation between fold maps and immersions via cobordisms. In [12,13], we defined these invariants for negative codimensional 1 fold maps in full generality, and we used them to detect direct summands of the cobordism groups of negative codimensional fold maps. In this paper, we study these invariants in the case of -1 codimensional fold maps with some additional restrictions about their singular fibers (for singular fibers, see [17,27]).
Simple fold maps are fold maps with at most one singular point in each connected component of a singular fiber. From this definition it follows that the only possible singular fibers whose singular points have sign “-” in the above normal form are the disjoint unions of a finite number of “figure eight” singular fibers and circle components, provided that the source manifold is orientable or the simple fold map is oriented (there is a consistent orientation of all fibers at their regular points). Simple fold maps have been studied, for example, by Levine [17], Saeki [23,24,25], Sakuma [29] and Yonebayashi [38]. The existence of a simple fold map on a manifold gives strong conditions about the structure of the manifold (for example, see the existence of simple fold maps on orientable 3-manifolds [25]). If we have a simple fold map of an oriented manifold or an oriented simple fold map, then the immersion of the singular set has trivial normal bundle in the target manifold N n , moreover there is a canonical trivialization corresponding to the number of regular fiber components in a neighbourhood of a singular fiber.
The main result of this paper is that our geometric invariants describe completely the set of cobordism classes of oriented simple fold maps of (n + 1)-dimensional manifolds into an n-dimensional manifold N n . By using Pontryagin-Thom type construction, we prove that the cobordism classes of oriented simple fold maps of (n + 1)-dimensional manifolds into an n-dimensional manifold N n are in a natural bijection with the set of stable homotopy classes of continuous maps of the one point compactification of the manifold N n into the Thom-space of the trivial line bundle over the space RP ∞ . As a special case, we obtain that the oriented cobordism group of simple fold maps of oriented (n + 1)-manifolds into R n is isomorphic to the nth stable homotopy group of the space S 1 ∨ SRP ∞ . We also describe the natural homomorphism which maps a simple fold cobordism class to its fold cobordism class in terms of natural homomorphisms between cobordism groups of immersions with prescribed normal bundles. In this way, we obtain results about the “inclusion” of the simple fold maps into the cobordism group of fold maps. We have the analogous results about bordisms (see Definition 5.1) of fold maps as well.
The paper is organized as follows. In Section 1 we give several basic definitions and notations. In Section 2 we state our main results. In Section 3 we prove our main theorems. In Section 4 we give explicit descriptions of the “inclusion” of the simple fold cobordism groups into the fold cobordism groups in low dimensions. In Section 5 we give analogous results about bordism groups of simple fold maps. In Appendix we prove theorems about the singular fibers of simple fold maps.
The author would like to thank Prof. András Szűcs for the uncountable discussions and suggestions, and Prof. Osamu Saeki for giving useful observations and correcting the appearance of this paper.
Notations. In this paper the symbol “∐” denotes the disjoint union, γ k denotes the universal k -dimensional real vector bundle over BO(k), ε k X (or ε k ) denotes the trivial k -plane bundle over the space X (resp. over the point), and the symbols ξ k , η k , etc. usually denote k -dimensional real vector bundles. The symbols detξ k and T ξ k denote the determinant line bundle and the Thom space of the bundle ξ k , respectively. The symbol Imm ξ k N (n -k, k) denotes the cobordism group of k -codimensional immersions into an n-dimensional manifold N whose normal bundles are induced from ξ k (this group is iso
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