Quantitative null-cobordism
HTML articles powered by AMS MathViewer
- by Gregory R. Chambers, Dominic Dotterrer, Fedor Manin and Shmuel Weinberger; with an appendix by Fedor Manin; with an appendix by Shmuel Weinberger
- J. Amer. Math. Soc. 31 (2018), 1165-1203
- DOI: https://doi.org/10.1090/jams/903
- Published electronically: July 10, 2018
- HTML | PDF | Request permission
Abstract:
For a given null-cobordant Riemannian $n$-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on $n$. In the appendix the bound is improved to one that is $O(L^{1+\varepsilon })$ for every $\varepsilon >0$.
This construction relies on another of independent interest. Take $X$ and $Y$ to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose $Y$ is simply connected and rationally homotopy equivalent to a product of Eilenberg–MacLane spaces, for example, any simply connected Lie group. Then two homotopic $L$-Lipschitz maps $f,g:X \to Y$ are homotopic via a $CL$-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces $Y$.
References
- Herbert Amann, Uniformly regular and singular Riemannian manifolds, Elliptic and parabolic equations, Springer Proc. Math. Stat., vol. 119, Springer, Cham, 2015, pp. 1–43. MR 3375165, DOI 10.1007/978-3-319-12547-3_{1}
- Hans J. Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Mathematics, Vol. 628, Springer-Verlag, Berlin-New York, 1977. MR 0467748, DOI 10.1007/BFb0065144
- Jean-Daniel Boissonnat, Ramsay Dyer, and Arijit Ghosh, Delaunay triangulation of manifolds, Found. Comput. Math. 18 (2018), no. 2, 399–431. MR 3777784, DOI 10.1007/s10208-017-9344-1
- Sandro Buoncristiano and Derek Hacon, An elementary geometric proof of two theorems of Thom, Topology 20 (1981), no. 1, 97–99. MR 592572, DOI 10.1016/0040-9383(81)90016-1
- Gregory R. Chambers, Fedor Manin, and Shmuel Weinberger, Quantitative nullhomotopy and rational homotopy type, Geom. Funct. Anal. 28 (2018), no. 3, 563–588. MR 3816519, DOI 10.1007/s00039-018-0450-2
- Jeff Cheeger and Mikhael Gromov, On the characteristic numbers of complete manifolds of bounded curvature and finite volume, Differential geometry and complex analysis, Springer, Berlin, 1985, pp. 115–154. MR 780040
- Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
- Francesco Costantino and Dylan Thurston, 3-manifolds efficiently bound 4-manifolds, J. Topol. 1 (2008), no. 3, 703–745. MR 2417451, DOI 10.1112/jtopol/jtn017
- Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, Simplicial matrix-tree theorems, Trans. Amer. Math. Soc. 361 (2009), no. 11, 6073–6114. MR 2529925, DOI 10.1090/S0002-9947-09-04898-3
- Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, Cellular spanning trees and Laplacians of cubical complexes, Adv. in Appl. Math. 46 (2011), no. 1-4, 247–274. MR 2794024, DOI 10.1016/j.aam.2010.05.005
- Marcelo Disconzi, Yuanzhen Shao, and Gieri Simonett, Some remarks on uniformly regular Riemannian manifolds, Math. Nachr. 289 (2016), no. 2-3, 232–242. MR 3458304, DOI 10.1002/mana.201400354
- H. Edelsbrunner and D. R. Grayson, Edgewise subdivision of a simplex, Discrete Comput. Geom. 24 (2000), no. 4, 707–719. ACM Symposium on Computational Geometry (Miami, FL, 1999). MR 1799608, DOI 10.1145/304893.304897
- Jürgen Eichhorn, The boundedness of connection coefficients and their derivatives, Math. Nachr. 152 (1991), 145–158. MR 1121230, DOI 10.1002/mana.19911520113
- Shai Evra and Tali Kaufman, Bounded degree cosystolic expanders of every dimension, STOC’16—Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, ACM, New York, 2016, pp. 36–48. MR 3536553, DOI 10.1145/2897518.2897543
- David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston, Word processing in groups, Jones and Bartlett Publishers, Boston, MA, 1992. MR 1161694, DOI 10.1201/9781439865699
- Herbert Federer and Wendell H. Fleming, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. MR 123260, DOI 10.2307/1970227
- Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
- Steve Ferry and Shmuel Weinberger, Quantitative algebraic topology and Lipschitz homotopy, Proc. Natl. Acad. Sci. USA 110 (2013), no. 48, 19246–19250. MR 3153953, DOI 10.1073/pnas.1208041110
- Misha Gromov and Larry Guth, Generalizations of the Kolmogorov-Barzdin embedding estimates, Duke Math. J. 161 (2012), no. 13, 2549–2603. MR 2988903, DOI 10.1215/00127094-1812840
- Phillip A. Griffiths and John W. Morgan, Rational homotopy theory and differential forms, Progress in Mathematics, vol. 16, Birkhäuser, Boston, Mass., 1981. MR 641551
- M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21st century, Vol. II (New Brunswick, NJ, 1993) Progr. Math., vol. 132, Birkhäuser Boston, Boston, MA, 1996, pp. 1–213. MR 1389019, DOI 10.1007/s10107-010-0354-x
- M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, vol. 152, Birkhäuser Boston, 1998.
- M. Gromov, Quantitative homotopy theory, Invited Talks on the Occasion of the 250th Anniversary of Princeton University (H. Rossi, ed.), Prospects in Mathematics, 1999, pp. 45–49.
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362, DOI 10.1007/978-1-4684-9449-5
- Stéfan Hildebrandt, Helmut Kaul, and Kjell-Ove Widman, An existence theorem for harmonic mappings of Riemannian manifolds, Acta Math. 138 (1977), no. 1-2, 1–16. MR 433502, DOI 10.1007/BF02392311
- Gil Kalai, Enumeration of $\textbf {Q}$-acyclic simplicial complexes, Israel J. Math. 45 (1983), no. 4, 337–351. MR 720308, DOI 10.1007/BF02804017
- Stephan Klaus and Matthias Kreck, A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres, Math. Proc. Cambridge Philos. Soc. 136 (2004), no. 3, 617–623. MR 2055050, DOI 10.1017/S0305004103007114
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
- Stefan Peters, Cheeger’s finiteness theorem for diffeomorphism classes of Riemannian manifolds, J. Reine Angew. Math. 349 (1984), 77–82. MR 743966, DOI 10.1515/crll.1984.349.77
- C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 69, Springer-Verlag, New York-Heidelberg, 1972. MR 0350744, DOI 10.1007/978-3-642-81735-9
- Thomas Schick, Manifolds with boundary and of bounded geometry, Math. Nachr. 223 (2001), 103–120. MR 1817852, DOI 10.1002/1522-2616(200103)223:1<103::AID-MANA103>3.3.CO;2-J
- Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554, DOI 10.1007/978-1-4684-9322-1
- Dennis Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1–79. MR 442930, DOI 10.2307/1970841
Bibliographic Information
- Gregory R. Chambers
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
- MR Author ID: 1075973
- Email: gchambers@rice.edu
- Dominic Dotterrer
- Affiliation: Department of Computer Science, Stanford University, Stanford, California 94305
- MR Author ID: 1008848
- Email: dominicd@cs.stanford.edu
- Fedor Manin
- Affiliation: Department of Mathematics, Ohio State University, Columbus, Ohio 43210
- MR Author ID: 1165933
- ORCID: 0000-0002-4545-6998
- Email: manin@math.toronto.edu
- Shmuel Weinberger
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 181430
- Email: shmuel@math.uchicago.edu
- Received by editor(s): December 21, 2016
- Received by editor(s) in revised form: July 24, 2017, January 21, 2018, and May 18, 2018
- Published electronically: July 10, 2018
- Additional Notes: The first author was partially supported by NSERC Postdoctoral Fellowship PDF-487617-2016.
The fourth author was partially supported by NSF grant DMS-1510178. - © Copyright 2018 American Mathematical Society
- Journal: J. Amer. Math. Soc. 31 (2018), 1165-1203
- MSC (2010): Primary 53C23; Secondary 57R75
- DOI: https://doi.org/10.1090/jams/903
- MathSciNet review: 3836564