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Discrete Morse theory and classifying spaces

Nanda, Vidit Tamaki, Dai Tanaka, Kohei 信州大学 DOI:10.1016/j.aim.2018.10.016

2021.02.16

概要

The aim of this paper is to develop a refinement of Forman's discrete Morse theory. To an acyclic partial matching mu on a finite regular CW complex X, Forman introduced a discrete analogue of gradient flows. Although Forman's gradient flow has been proved to be useful in practical computations of homology groups, it is not sufficient to recover the homotopy type of X. Forman also proved the existence of a CW complex which is homotopy equivalent to X and whose cells are in one-to-one correspondence with the critical cells of mu but the construction is ad hoc and does not have a combinatorial description. By relaxing the definition of Forman's gradient flows, we introduce the notion of flow paths, which contains enough information to reconstruct the homotopy type of X, while retaining a combinatorial description. The critical difference from Forman's gradient flows is the existence of a partial order on the set of flow paths, from which a 2-category C(mu) is constructed. It is shown that the classifying space of C(mu) is homotopy equivalent to X by using homotopy theory of 2-categories. This result can be also regarded as a discrete analogue of the unpublished work of Cohen, Jones, and Segal on Morse theory in early 90's. (C) 2018 Elsevier Inc. All rights reserved.

参考文献

[BBL+ 99] Eric Babson, Anders Bj¨

orner, Svante Linusson, John Shareshian, and Volkmar

Welker. Complexes of not i-connected graphs. Topology, 38(2):271–299, 1999,

arXiv:math/9705219.

[BBM07] Mladen Bestvina, Kai-Uwe Bux, and Dan Margalit. Dimension of the Torelli group

for Out(Fn ). Invent. Math., 170(1):1–32, 2007, arXiv:math/0603177.

[BC03]

M. Bullejos and A. M. Cegarra. On the geometry of 2-categories and their classifying spaces. K-Theory, 29(3):211–229, 2003, http://dx.doi.org/10.1023/B:

KTHE.0000006921.50151.00.

[Bj¨

o95]

A. Bj¨

orner. Topological methods. In Handbook of combinatorics, Vol. 1, 2, pages

1819–1872. Elsevier, Amsterdam, 1995.

[Bro60]

Morton Brown. A proof of the generalized Schoenflies theorem. Bull. Amer. Math.

Soc., 66:74–76, 1960.

[BW02]

E. Batzies and V. Welker. Discrete Morse theory for cellular resolutions. J. Reine

Angew. Math., 543:147–168, 2002, http://dx.doi.org/10.1515/crll.2002.012.

[CCG10] Pilar Carrasco, Antonio M. Cegarra, and Antonio R. Garz´on. Nerves and classifying

spaces for bicategories. Algebr. Geom. Topol., 10(1):219–274, 2010, arXiv:0903.5058.

53

[Ceg11]

A. M. Cegarra. Homotopy fiber sequences induced by 2-functors. J. Pure Appl.

Algebra, 215(4):310–334, 2011, arXiv:0909.4229.

[CJS]

R. L. Cohen, J.D.S. Jones, and G. B. Segal. Morse theory and classifying spaces,

http://math.stanford.edu/~ralph/morse.ps. preprint.

[CJS94]

R. L. Cohen, J. D. S. Jones, and G. B. Segal. Floer’s infinite-dimensional Morse theory

and homotopy theory. S¯

urikaisekikenky¯

usho K¯

oky¯

uroku, (883):68–96, 1994, http://

www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/0883-4.pdf. Geometric aspects of infinite integrable systems (Japanese) (Kyoto, 1993).

[CJS95]

R. L. Cohen, J. D. S. Jones, and G. B. Segal. Floer’s infinite-dimensional Morse

theory and homotopy theory. In The Floer memorial volume, volume 133 of Progr.

Math., pages 297–325. Birkh¨auser, Basel, 1995.

[Coh73]

Marshall M. Cohen. A course in simple-homotopy theory. Springer-Verlag, New YorkBerlin, 1973. Graduate Texts in Mathematics, Vol. 10.

[Del08]

Emanuele Delucchi. Shelling-type orderings of regular CW-complexes and acyclic

matchings of the Salvetti complex. Int. Math. Res. Not. IMRN, (6):Art. ID rnm167,

39, 2008, arXiv:0705.3107.

[dH12]

Matias L. del Hoyo. On the loop space of a 2-category. J. Pure Appl. Algebra,

216(1):28–40, 2012, arXiv:1005.1300.

[DS10]

Emanuele Delucchi and Simona Settepanella. Combinatorial polar orderings and

recursively orderable arrangements. Adv. in Appl. Math., 44(2):124–144, 2010,

arXiv:0711.1517.

[Dug08]

Danial Dugger. A primer on homotopy colimits, 2008, http://math.uoregon.edu/

~ddugger/hocolim.pdf.

[Dus02]

John W. Duskin. Simplicial matrices and the nerves of weak n-categories. I. Nerves of

bicategories. Theory Appl. Categ., 9:198–308, 2001/02. CT2000 Conference (Como).

[For95]

Robin Forman. A discrete Morse theory for cell complexes. In Geometry, topology,

& physics, Conf. Proc. Lecture Notes Geom. Topology, IV, pages 112–125. Int. Press,

Cambridge, MA, 1995.

[For98]

Robin Forman. Morse theory for cell complexes. Adv. Math., 134(1):90–145, 1998,

http://dx.doi.org/10.1006/aima.1997.1650.

[Fra79]

John M. Franks. Morse-Smale flows and homotopy theory. Topology, 18(3):199–215,

1979, http://dx.doi.org/10.1016/0040-9383(79)90003-X.

[Gal10]

Etienne

Gallais. Combinatorial realization of the Thom-Smale complex via discrete Morse theory. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 9(2):229–252, 2010,

arXiv:0803.2616.

[Gra80]

John W. Gray. Closed categories, lax limits and homotopy limits. J. Pure Appl.

Algebra, 19:127–158, 1980, http://dx.doi.org/10.1016/0022-4049(80)90098-5.

[JW09]

Michael J¨

ollenbeck and Volkmar Welker. Minimal resolutions via algebraic discrete

Morse theory. Mem. Amer. Math. Soc., 197(923):vi+74, 2009.

54

[Kal75]

Gudrun Kalmbach. On some results in Morse theory. Canad. J. Math., 27:88–105,

1975.

[Kas95]

Christian Kassel. Quantum groups, volume 155 of Graduate Texts in Mathematics.

Springer-Verlag, New York, 1995, http://dx.doi.org/10.1007/

978-1-4612-0783-2.

[Koz05]

Dmitry N. Kozlov. Discrete Morse theory for free chain complexes. C. R. Math. Acad.

Sci. Paris, 340(12):867–872, 2005, arXiv:cs.DM/0504090.

[Koz08]

Dmitry Kozlov. Combinatorial algebraic topology, volume 21 of Algorithms and Computation in Mathematics. Springer, Berlin, 2008, http://dx.doi.org/10.1007/

978-3-540-71962-5.

[Lac10]

Stephen Lack. A 2-categories companion. In Towards higher categories, volume 152 of IMA Vol. Math. Appl., pages 105–191. Springer, New York, 2010,

arXiv:math/0702535.

[LW69]

Albert T. Lundell and Stephen Weingram. Topology of CW-Complexes. Van Nostrand

Reinhold, New York, 1969.

[May72]

J. P. May. The geometry of iterated loop spaces. Springer-Verlag, Berlin, 1972.

[ML98]

Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate

Texts in Mathematics. Springer-Verlag, New York, 1998.

[MN13]

Konstantin Mischaikow and Vidit Nanda. Morse Theory for Filtrations and Efficient

Computation of Persistent Homology. Discrete Comput. Geom., 50(2):330–353, 2013,

http://dx.doi.org/10.1007/s00454-013-9529-6.

[Qui73]

Daniel Quillen. Higher algebraic K-theory. I. In Algebraic K-theory, I: Higher Ktheories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), pages 85–147.

Lecture Notes in Math., Vol. 341. Springer, Berlin, 1973.

[RS71]

C. P. Rourke and B. J. Sanderson. 4-sets. I. Homotopy theory. Quart. J. Math.

Oxford Ser. (2), 22:321–338, 1971.

[Rus73]

T. Benny Rushing. Topological embeddings. Academic Press, New York-London, 1973.

Pure and Applied Mathematics, Vol. 52.

[Seg68]

Graeme Segal. Classifying spaces and spectral sequences. Inst. Hautes Etudes

Sci.

Publ. Math., 34:105–112, 1968.

[Seg73]

Graeme Segal. Configuration-spaces and iterated loop-spaces. Invent. Math., 21:213–

221, 1973.

[Seg74]

Graeme Segal. Categories and cohomology theories. Topology, 13:293–312, 1974.

[Sha01]

John Shareshian. Discrete Morse theory for complexes of 2-connected graphs. Topology, 40(4):681–701, 2001, http://dx.doi.org/10.1016/S0040-9383(99)00076-2.

[Sk¨

o06]

Emil Sk¨

oldberg.

Morse theory from an algebraic viewpoint.

Trans. Amer.

Math. Soc., 358(1):115–129 (electronic), 2006, http://dx.doi.org/10.1090/

S0002-9947-05-04079-1.

55

[SS07]

Mario Salvetti and Simona Settepanella. Combinatorial Morse theory and minimality

of hyperplane arrangements. Geom. Topol., 11:1733–1766, 2007, arXiv:0705.2874.

[Str96]

Ross Street. Categorical structures. In Handbook of algebra, Vol. 1, volume 1 of

Handb. Algebr., pages 529–577. Elsevier/North-Holland, Amsterdam, 1996, http:

//dx.doi.org/10.1016/S1570-7954(96)80019-2.

[Wat66]

Charles E. Watts. A homology theory for small categories. In Proc. Conf. Categorical

Algebra (La Jolla, Calif., 1965), pages 331–335. Springer, New York, 1966.

56

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