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Indefinite Morse 2-functions: broken fibrations and generalizations. (English) Zbl 1328.57019

A Morse 2-function is a generic smooth map from a manifold \(M^n\) to a surface, like classical Morse functions are generic smooth maps from \(M^n\) to \({\mathbb R}^1\). The singularities of 2-Morse functions are folds and cusps, where folds look locally like \((t, x_1,\ldots,x_{n-1})\to (t,f(x_1,\ldots,x_{n-1}))\) for a standard Morse singularity \(f\), and cusps look locally like \((t; x_1,\ldots,x_{n-1})\to(t, f_t(x_1,\ldots,x_{n-1}))\) for a standard birth \(f_t\) of a cancelling pair of Morse singularities.
The paper under review develops techniques for working with Morse 2-functions and generic homotopies between them, paying particular attention to avoiding definite folds, in which the modeling function \(f\) is a local extremum, and to guaranteeing connected fibers. When definite folds are avoided, the Morse 2-function (or generic homotopy) is said to be indefinite.
The main result is to extend the existence and uniqueness results to indefinite, Morse 2-functions with connected fibers. These results are proved for the general case that \(M^n\) is a manifold with boundary, mapping to a surface \(\Sigma^2\) with boundary. In the closed case the results specialize as follows.
For \(n>2\) a map \(f: M^n\to \Sigma^2\) from a closed \(n\)-manifold to a surface is homotopic to an indefinite Morse 2-function if and only if \(f_*(\pi_1M)\) has finite index in \(\pi_1\Sigma\). (The necessity of this condition is due to O. Saeki [Kyushu J. Math. 60, No. 2, 363–382 (2006; Zbl 1113.57016)]). If \(n>3\), then any such indefinite Morse 2-functions are homotopic through an indefinite generic homotopy.

MSC:

57M50 General geometric structures on low-dimensional manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension

Citations:

Zbl 1113.57016
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References:

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