Hiroshima Mathematical Journal
- Hiroshima Math. J.
- Volume 36, Number 1 (2006), 141-170.
Morse functions with sphere fibers
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.
Hiroshima Math. J., Volume 36, Number 1 (2006), 141-170.
First available in Project Euclid: 17 May 2006
Permanent link to this document
Digital Object Identifier
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
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