## Quantitative null-cobordism

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Gregory R. Chambers, Dominic Dotterrer, Fedor Manin and Shmuel Weinberger; with an appendix by Fedor Manin; with an appendix by Shmuel Weinberger
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## 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

## Additional 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