A Generalized Discrete Morse–Floer Theory

Abstract

Forman has developed a version of discrete Morse theory that can be understood in terms of arrow patterns on a (simplicial, polyhedral or cellular) complex without closed orbits, where each cell may either have no arrows, receive a single arrow from one of its facets, or conversely, send a single arrow into a cell of which it is a facet. By following arrows, one can then construct a natural Floer-type boundary operator. Here, we develop such a construction for arrow patterns where each cell may support several outgoing or incoming arrows (but not both), again in the absence of closed orbits. Our main technical achievement is the construction of a boundary operator that squares to 0 and therefore recovers the homology of the underlying complex.

Keywords

CW complex / Boundary operator / Floer theory / Poincaré polynomial / Betti number / Discrete Morse theory / Discrete Morse–Floer theory / Conley theory

References

[1] Akaho M. Morse homology and manifolds with boundary. Commun. Contemp. Math.. 2007, 9 3 301-334

[2] Banyaga A, Hurtubise D. Lectures on Morse Homology, Kluwer Texts in Mathematical Sciences. 2004 Dordrecht: Kluwer Academic Publishers

[3] Bott R. Nondegenerate critical manifolds. Ann. Math. (2). 1954, 60 248-261

[4] Bott R. Lectures on Morse theory, old and new. Bull. Am. Math. Soc. (N.S.). 1982, 7 2 331-358

[5] Bredon GE. Topology and Geometry, Graduate Texts in Mathematics. 1997 Berlin: Springer

[6] Conley, C.: Isolated invariant sets and the Morse index. In: CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society (1978)

[7] Corvellec J-N. Morse theory for continuous functionals. J. Math. Anal. Appl.. 1995, 196 03 1050-1072

[8] Dieck T. Algebraic Topology, EMS Textbooks in Mathematics. 2008 Zürich: European Mathematical Society (EMS)

[9] Floer A. Witten’s complex and infinite dimensional Morse theory. J. Differ. Geom.. 1989, 30 207-221

[10] Forman R. Morse theory for cell complexes. Adv. Math.. 1998, 134 90-145

[11] Forman R. Witten–Morse theory for cell complexes. Topology. 1998, 37 945-979

[12] Forman R. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift. 1998, 228 4 629-681

[13] Fritsch R, Piccinini RA. Cellular Structures in Topology. 1990 Cambridge: Cambridge University Press

[14] Geoghegan R. Topological Methods in Group Theory, Graduate Texts in Mathematics. 2008 New York: Springer

[15] Hatcher A. Algebraic Topology. 2002 Cambridge: Cambridge University Press

[16] Horn RA, Johnson CR. Matrix Analysis. 1985 Cambridge: Cambridge University Press

[17] Jost J. Riemannian Geometry and Geometric Analysis. 2017 7 New York: Springer

[18] Jost J. Dynamical Systems, Universitext. 2005 Berlin: Springer

[19] Kozlov D. Combinatorial Algebraic Topology. 2008 New York: Springer

[20] Lundell AT, Weingram S. The Topology of CW Complexes. 1969 New York: Van Nostrand

[21] Milnor J. Morse Theory, Annals of Mathematics Studies. 1963 Princeton: Princeton University Press

[22] Mischaikow K. Conley Index Theory: A Brief Introduction. 1999 Warsaw: Banach Center Publications

[23] Morse M. Relations between the critical points of a real function of $n$ independent variables. Trans. Am. Math. Soc. 1925, 27 345-396

[24] Munkres JR. Elements of Algebraic Topology. 1984 Reading: Addison-Wesley

[25] Schwarz M. Morse Homology. 1993 Basel: Birkhäuser

[26] Whitehead JCH. Combinatorial homotopy I. Bull. Am. Math. Soc.. 1949, 55 213-245

[27] Witten E. Supersymmetry and Morse theory. J. Differ. Geom.. 1982, 17 661-692

[28] Yaptieu, S.: Generalizations of discrete Morse theory, Thesis, Leipzig (2017)

[29] Yaptieu, S.: Discrete Morse–Bott theory, (MIS Preprint) (2017)

[30] Yaptieu, S.: Discrete Morse–Bott theory for CW complexes, MIS Preprint (2017)

Funding

IMPRS Stipend