Hiroshima Mathematical Journal

Morse functions with sphere fibers

Osamu Saeki

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A smooth closed manifold is said to be an almost sphere if it admits a Morse function with exactly two critical points. In this paper, we characterize those smooth closed manifolds which admit Morse functions such that each regular fiber is a finite disjoint union of almost spheres. We will see that such manifolds coincide with those which admit Morse functions with at most three critical values. As an application, we give a new proof of the characterization theorem of those closed manifolds which admit special generic maps into the plane. We also discuss homotopy and diffeomorphism invariants of manifolds related to the minimum number of critical values of Morse functions; in particular, the Lusternik-Schnirelmann category and spherical cone length. Those closed orientable 3-manifolds which admit Morse functions with regular fibers consisting of spheres and tori are also studied.

Article information

Hiroshima Math. J., Volume 36, Number 1 (2006), 141-170.

First available in Project Euclid: 17 May 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R65: Surgery and handlebodies
Secondary: 57R70: Critical points and critical submanifolds 57R60: Homotopy spheres, Poincaré conjecture 55M30: Ljusternik-Schnirelman (Lyusternik-Shnirelʹman) category of a space 57N10: Topology of general 3-manifolds [See also 57Mxx] 58K05: Critical points of functions and mappings

Morse function sphere fiber critical values special generic map Lusternik-Schnirelmann category homotopy sphere handlebody decomposition Heegaard genus


Saeki, Osamu. Morse functions with sphere fibers. Hiroshima Math. J. 36 (2006), no. 1, 141--170. doi:10.32917/hmj/1147883401. https://projecteuclid.org/euclid.hmj/1147883401

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