Fold Maps and Immersions from the Viewpoint of Cobordism
نویسنده
چکیده
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 for N = R the 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 space RP∞ . 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.
منابع مشابه
Fold Maps, Framed Immersions and Smooth Structures
We show that the cobordism group of fold maps of even non-positive codimension q into a manifold N is a sum of q/2 cobordism groups of framed immersions into N and a group related to diffeomorphism groups of manifolds of dimension q + 1. In the case of maps of odd non-positive codimension q, we show that the cobordism groups of fold maps split off (q − 1)/2 cobordism groups of framed immersions.
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We give complete geometric invariants of cobordisms of fold maps with oriented singular set and cobordisms of even codimensional fold maps. These invariants are given in terms of cobordisms of stably framed manifolds and cobordisms of immersions with prescribed normal bundles.
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We compute the oriented cobordism group of fold maps of 4-manifolds into R 3 with all the possible restrictions (and also with no restriction) to the singular fibers. We also give geometric invariants which describe completely the cobordism group of fold maps.
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A generic smooth map of a closed 2k-manifold into (3k− 1)-space has a finite number of cusps (Σ-singularities). We determine the possible numbers of cusps of such maps. A fold map is a map with singular set consisting of only fold singularities (Σsingularities). Two fold maps are fold bordant if there are cobordisms between their sourceand target manifolds with a fold map extending the two maps...
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