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Remarks on the topology of folds. (English) Zbl 0828.57025

We give some necessary conditions for a closed manifold to admit a smooth map into a Euclidean space with only fold singular points. The main results of this paper are as follows:
Theorem 1.1. If a closed \(n\)-dimensional manifold \(M^n\) admits a smooth map \(f : M^n \to \mathbb{R}^p\) \((n \geq p)\) with only definite fold singular points, then it is smoothly null-cobordant. If, in addition, \(M^n\) is oriented, it is null-cobordant in the oriented category.
Theorem 1.2. Let \(M^n\) be a closed \(n\)-dimensional manifold with odd Euler number. If \(M^n\) admits a smooth map \(f : M^n \to \mathbb{R}^p\) \((n \geq p)\) with only fold singular points, then \(p = 1\), 3, or 7.
Note that Theorem 1.1 has been proved in [the second author, Topology Appl. 49, No. 3, 265-293 (1993; Zbl 0768.57015), Corollary 3.3] when \(n > p\).

MSC:

57R45 Singularities of differentiable mappings in differential topology
57R20 Characteristic classes and numbers in differential topology
57R42 Immersions in differential topology
57R75 \(\mathrm{O}\)- and \(\mathrm{SO}\)-cobordism
57R19 Algebraic topology on manifolds and differential topology

Citations:

Zbl 0768.57015
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References:

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