Journal of Mathematics of Kyoto University

Fold-maps and the space of base point preserving maps of spheres

Yoshifumi Ando

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Let $f : N \to P$ be a smooth map between $n$-dimensional oriented manifolds which has only fold singularities. Such a map is called a fold-map. For a connected closed oriented manifold $P$, we shall define a fold-cobordism class of a fold-map into $P$ of degree m under a certain cobordism equivalence. Let $\Omega _{fold,m}(P)$ denote the set of all foldcobordism classes of fold-maps into $P$ of degree $m$. Let $F^{m}$ denote the space $\lim _{k\to \infty}F_{k}^{m}$, where $F_{k}^{m}$ denotes the space of all base point preserving maps of degree $m$ of $S^{k-1}$. In this paper we shall prove that there exists a surjection of $\Omega _{fold,m}(P)$ to the set of homotopy classes $[P,F^{m}]$, which induces many fold-cobordism invariants.

Article information

J. Math. Kyoto Univ., Volume 41, Number 4 (2001), 693-737.

First available in Project Euclid: 17 August 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R90: Other types of cobordism [See also 55N22]
Secondary: 57R45: Singularities of differentiable mappings 58K30: Global theory


Ando, Yoshifumi. Fold-maps and the space of base point preserving maps of spheres. J. Math. Kyoto Univ. 41 (2001), no. 4, 693--737. doi:10.1215/kjm/1250517595.

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