Algebraic Topology and Related Topics pp 287-313 | Cite as
Stellar Stratifications on Classifying Spaces
Abstract
We extend Björner’s characterization of the face poset of finite CW complexes to a certain class of stratified spaces, called cylindrically normal stellar complexes. As a direct consequence, we obtain a discrete analogue of cell decompositions in smooth Morse theory, by using the classifying space model introduced in Nanda et al (Discrete Morse theory and classifying spaces, arXiv:1612.08429 [15]). As another application, we show that the exit-path category \(\mathsf {Exit}(X)\), in the sense of Lurie (Higher algebra, http://www.math.harvard.edu/~lurie/papers/HA.pdf [11]), of a finite cylindrically normal CW stellar complex X is a quasi-category.
Notes
Acknowledgements
This project started when the authors were invited to the IBS Center for Geometry and Physics in Pohang in December, 2016. We would like to thank the center for invitation and the nice working environment.
The contents of this paper were presented by the first author during the 7th East Asian Conference on Algebraic Topology held at Mohali, India, in December, 2017. He is grateful to the local organizers for the invitation to the conference and the hospitality of IISER Mohali.
The authors would like to thank the anonymous referee whose valuable suggestions improved expositions and made this paper more readable.
References
- 1.D. Ayala, J. Francis, H.L. Tanaka, Factorization homology of stratified spaces. Sel. Math. (N.S.) 23(1), 293–362 (2017), arXiv:1409.0848MathSciNetCrossRefGoogle Scholar
- 2.D. Ayala, J. Francis, H.L. Tanaka, Local structures on stratified spaces. Adv. Math. 307, 903–1028 (2017), arXiv:1409.0501MathSciNetCrossRefGoogle Scholar
- 3.R. Andrade, From manifolds to invariants of \({E}_n\)-algebras. ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–Massachusetts Institute of TechnologyGoogle Scholar
- 4.A. Björner, Posets, regular CW complexes and Bruhat order. Eur. J. Combin. 5(1), 7–16 (1984). https://doi.org/10.1016/S0195-6698(84)80012-8MathSciNetCrossRefGoogle Scholar
- 5.R.L. Cohen, J.D.S. Jones, G.B. Segal, Morse theory and classifying spaces, http://math.stanford.edu/~ralph/morse.ps
- 6.C. de Seguins Pazzis, The geometric realization of a simplicial Hausdorff space is Hausdorff. Topology Appl. 160(13), 1621–1632 (2013), arXiv:1005.2666
- 7.J. Ebert, O. Randal-Williams, Semi-simplicial spaces. Algebr. Geom. Topol. arXiv:1705.03774
- 8.R. Forman, A discrete Morse theory for cell complexes, Geometry, Topology, and Physics (Conference Proceedings). Lecture Notes Geometry Topology, vol. 4 (International Press, Cambridge, 1995), pp. 112–125Google Scholar
- 9.R. Forman, Morse theory for cell complexes. Adv. Math. 134(1), 90–145 (1998). https://doi.org/10.1006/aima.1997.1650MathSciNetCrossRefGoogle Scholar
- 10.A. Kirillov Jr., On piecewise linear cell decompositions. Algebr. Geom. Topol. 12(1), 95–108 (2012), arXiv:1009.4227
- 11.J. Lurie, Higher algebra, http://www.math.harvard.edu/~lurie/papers/HA.pdf
- 12.J. Lurie, Higher Topos Theory. Annals of Mathematics Studies, vol. 170 (Princeton University, Princeton, 2009). https://doi.org/10.1515/9781400830558
- 13.A.T. Lundell, S. Weingram, Topology of CW-Complexes (Van Nostrand Reinhold, New York, 1969)Google Scholar
- 14.J. Mather, Notes on Topological Stability (Harvard University, Cambridge, 1970)Google Scholar
- 15.V. Nanda, D. Tamaki, K. Tanaka, Discrete Morse theory and classifying spaces. Adv. Math. 340, 723–790 (2018), arXiv:1612.08429MathSciNetCrossRefGoogle Scholar
- 16.D. Tamaki, Cellular stratified spaces, Combinatorial and Toric Homotopy: Introductory Lectures. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, vol. 35 (World Scientific, Singapore, 2018), pp. 305–453, arXiv:1609.04500Google Scholar
- 17.R. Thom, Ensembles et morphismes stratifiés. Bull. Am. Math. Soc. 75, 240–284 (1969). https://doi.org/10.1090/S0002-9904-1969-12138-5MathSciNetCrossRefGoogle Scholar