Journal of the Mathematical Society of Japan

Folding maps and the surgery theory on manifolds

Yoshifumi ANDO

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Abstract

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

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

Dates
First available in Project Euclid: 9 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213023462

Digital Object Identifier
doi:10.2969/jmsj/05320357

Mathematical Reviews number (MathSciNet)
MR1815139

Zentralblatt MATH identifier
0980.58026

Subjects
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

Keywords
Folding singularity jet space manifold surgery theory homotopy class

Citation

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. https://projecteuclid.org/euclid.jmsj/1213023462


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