Morse theory for cell complexes.(English)Zbl 0896.57023
This paper provides a discrete Morse theory for CW-complexes. The author proves an analogue of the main theorems of the classical theory and gets in the context of PL manifolds an interesting formula of the PL Poincaré conjecture.
“The following are equivalent: (1) (PL Poincaré conjecture) Let \(M\) be a PL \(n\)-manifold which is a homotopy \(n\)-sphere. Then \(M\) is a PL \(n\)-sphere; (2) Let \(M\) be a PL \(n\)-manifold which is a homotopy \(n\)-sphere. Then, by a series of bisections, \(M\) can be subdivided to a complex which has a Morse function with exactly 2 critical points.”
The author builds also the gradient vector field of a discrete (or combinatorial) Morse function and an associated differential complex with the same homology as the underlying manifold. This discrete Morse theory can then be used to give a Morse theoretic proof à la Milnor of the PL \(s\)-cobordism theorem [J. W. Milnor, Lectures on the \(h\)-cobordism theorem (1965;Zbl 0161.20302)].
Main of the final part of the paper is devoted to characterize gradient vector fields of discete Morse functions. Doing that, the author shows that every discrete Morse function can be replaced by a self-indexing one, with the same critical points.
In spite of some misprints or change without notice (e.g. combinatorial for discrete) this paper is pleasant and not too hard to read; there are many examples. It should become a reference in the subject.
“The following are equivalent: (1) (PL Poincaré conjecture) Let \(M\) be a PL \(n\)-manifold which is a homotopy \(n\)-sphere. Then \(M\) is a PL \(n\)-sphere; (2) Let \(M\) be a PL \(n\)-manifold which is a homotopy \(n\)-sphere. Then, by a series of bisections, \(M\) can be subdivided to a complex which has a Morse function with exactly 2 critical points.”
The author builds also the gradient vector field of a discrete (or combinatorial) Morse function and an associated differential complex with the same homology as the underlying manifold. This discrete Morse theory can then be used to give a Morse theoretic proof à la Milnor of the PL \(s\)-cobordism theorem [J. W. Milnor, Lectures on the \(h\)-cobordism theorem (1965;Zbl 0161.20302)].
Main of the final part of the paper is devoted to characterize gradient vector fields of discete Morse functions. Doing that, the author shows that every discrete Morse function can be replaced by a self-indexing one, with the same critical points.
In spite of some misprints or change without notice (e.g. combinatorial for discrete) this paper is pleasant and not too hard to read; there are many examples. It should become a reference in the subject.
Reviewer: R.Barre (Valenciennes)
MSC:
| 57R70 | Critical points and critical submanifolds in differential topology |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
| 57R80 | \(h\)- and \(s\)-cobordism |
| 57R60 | Homotopy spheres, Poincaré conjecture |
| 57Q99 | PL-topology |
Citations:
Zbl 0161.20302Cite
References:
| [1] | Banchoff, T., Critical points and curvature for embedded polyhedra, J. Diff. Geo., 1, 245-256 (1967) ·Zbl 0164.22903 |
| [2] | Barden, D., The Structure of Manifolds (1963), Cambridge University ·Zbl 0136.20602 |
| [3] | Brehm, U.; Kühnel, W., Combinatorial manifolds with few vertices, Topology, 26, 465-473 (1987) ·Zbl 0681.57009 |
| [4] | A. Duval, A combinatorial decomposition of Simplicial complexes, Israel J. Math.; A. Duval, A combinatorial decomposition of Simplicial complexes, Israel J. Math. ·Zbl 0820.55001 |
| [5] | Eells, J.; Kuiper, N., Manifolds which are like projective planes, Inst. Hautes Etudes Sci. Publ. Math. No. 14, 5-46 (1962) ·Zbl 0109.15701 |
| [6] | Forman, R., Determinants of Laplacians on graphs, Topology, 32, 35-46 (1993) ·Zbl 0780.05041 |
| [7] | R. Forman, Witten-Morse theory for cell complexes, Topology; R. Forman, Witten-Morse theory for cell complexes, Topology ·Zbl 0929.57020 |
| [8] | Forman, R., A discrete Morse theory for cell complexes, Geometry, Topology & Physics for Raoul Bott (1995), International Press ·Zbl 0867.57018 |
| [9] | R. Forman, Combinatorial vector fields and dynamical systems, Math. Zeit.; R. Forman, Combinatorial vector fields and dynamical systems, Math. Zeit. ·Zbl 0922.58063 |
| [10] | Glaser, L. C., Geometrical Cominatorial Topology (1970), Van Nostrand-Reinhold: Van Nostrand-Reinhold New York ·Zbl 0212.55603 |
| [11] | Goresky, M.; MacPherson, R., Stratified Morse Theory. Stratified Morse Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 14 (1988), Springer-Verlag: Springer-Verlag Berlin ·Zbl 0639.14012 |
| [12] | Klingenberg, W., The Morse complex, Symposia Mathematica Vol. XXVI (Rome, 1980) (1982), Academic Press: Academic Press New York, p. 117-122 |
| [13] | Kosinski, A., Singularities of piecewise linear mappings. I. Mappings into the real line, Bull. Amer. Math. Soc., 68, 110-114 (1962) ·Zbl 0126.18801 |
| [14] | Kuhnel, W., Triangulations of manifolds with few vertices, Advances in Differential Geometry and Topology (1990), World Scientific: World Scientific Singapore, p. 59-114 ·Zbl 0836.57002 |
| [15] | Lundell, A.; Weingram, S., The Topology of CW Complexes (1969), Van Nostrand-Reinhold: Van Nostrand-Reinhold New York ·Zbl 0207.21704 |
| [16] | Mazur, B., Differential topology from the viewpoint of simple homotopy theory, Publ. Math. I.H.E.S., 15, 5-93 (1963) ·Zbl 0173.51203 |
| [17] | Milnor, J., Morse Theory, Annals of Mathematics Study, 51 (1962), Princeton Univ. Press: Princeton Univ. Press Princeton |
| [18] | Milnor, J., Lectures on the h-cobordism theorem, Princeton Mathematical Notes (1965), Princeton Univ. Press: Princeton Univ. Press Princeton ·Zbl 0161.20302 |
| [19] | Morse, M., Bowls of a non-degenerate function on a compact differentiable manifold, (Cairns, S. S., Differential and Combinatorial Topology (1965), Princeton Univ. Press: Princeton Univ. Press Princeton), 81-104 ·Zbl 0159.24803 |
| [20] | Smale, S., On gradient dynamical systems, Ann. Math., 74, 199-206 (1961) ·Zbl 0136.43702 |
| [21] | Smale, S., The generalized Poincaré conjecture in dimensions greater than four, Ann. Math., 74, 391-406 (1961) ·Zbl 0099.39202 |
| [22] | Stallings, J. R., Lectures on Polyhedral Topology (1968), Tata Institute of Fundamental Research: Tata Institute of Fundamental Research Bombay ·Zbl 0182.26203 |
| [23] | Stanley, R., A combinatorial decomposition of acyclic simplicial complexes, Discrete Math., 118 (1993) ·Zbl 0782.55004 |
| [24] | J. H. C. Whitehead, Simplicial spaces, nuclei and \(m\); J. H. C. Whitehead, Simplicial spaces, nuclei and \(m\) ·Zbl 0022.40702 |
| [25] | Witten, E., Supersymmetry and Morse theory, J. Differential Geom., 17, 661-692 (1982) ·Zbl 0499.53056 |
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