Journal of the Mathematical Society of Japan

Folding maps and the surgery theory on manifolds

Yoshifumi ANDO

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Let f:NP be a smooth map between n-dimensional oriented manifolds which has only folding singularities. Such a map is called a folding map. We prove that a folding map f : NP canonically determines the homotopy class of a bundle map of TNθN to TPθP, where θN and θP are the trivial line bundles over N and P respectively. When P is a closed manifold in addition, we define the set Ωfold(P) of all cobordism classes of folding maps of closed manifolds into P of degree 1 under a certain cobordism equivalence. Let SG denote the space limkSGk, where SGk denotes the space of all homotopy equivalences of Sk-1 of degree 1. We prove that there exists an important map of Ωfold(P) to the set of homotopy classes [P,SG]. We relate Ωfold(P) with the set of smooth structures on P by applying the surgery theory.

Article information

J. Math. Soc. Japan, Volume 53, Number 2 (2001), 357-382.

First available in Project Euclid: 9 June 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58K15: Topological properties of mappings
Secondary: 57R45: Singularities of differentiable mappings 57R67: Surgery obstructions, Wall groups [See also 19J25] 57R55: Differentiable structures 55Q10: Stable homotopy groups

Folding singularity jet space manifold surgery theory homotopy class


ANDO, Yoshifumi. Folding maps and the surgery theory on manifolds. J. Math. Soc. Japan 53 (2001), no. 2, 357--382. doi:10.2969/jmsj/05320357.

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