On macroscopic dimension of rationally inessential manifolds. (English. Russian original) Zbl 1271.53043

Funct. Anal. Appl. 45, No. 3, 187-191 (2011); translation from Funkts. Anal. Prilozh. 45, No. 3, 34-40 (2011).
Summary: We show that, for a rationally inessential orientable closed \(n\)-manifold \(M\) whose fundamental group is a duality group, the macroscopic dimension of its universal cover \(\widetilde M\) is strictly less than \(n:\dim_{MC}\widetilde M<n\). As a corollary, we obtain the following partial result towards Gromov’s conjecture:
The inequality \(\dim_{MC}\widetilde M<n\) holds for the universal cover \(\widetilde M\) of a closed spin \(n\)-manifold \(M\) with a positive scalar curvature metric if the fundamental group \(\pi_1(M)\) is a duality group satisfying the analytic Novikov conjecture.


53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Full Text: DOI


[1] V. I. Arnold, ”Dynamics of complexity of intersections,” Bol. Soc. Brasil. Mat. (N.S.), 21:1 (1990), 1–10. · Zbl 0782.54020
[2] D. Bolotov, ”Macroscopic dimension of 3-manifolds,” Math. Phys. Anal. Geom., 6 (2003), 291–299. · Zbl 1029.57016
[3] D. Bolotov and A. Dranishnikov, ”On Gromov’s scalar curvature conjecture,” Proc. Amer. Math. Soc., 138:4 (2010), 1517–1524. · Zbl 1193.53106
[4] S. Brendle and R. Schoen, ”Sphere theorems in geometry,” in: Surv. Differential Geometry, vol. XIII, Intern. Press, Somerville, MA, 2009, 49–84. · Zbl 1184.53037
[5] K. Brown, Cohomology of Groups, Springer-Verlag, New York-Berlin, 1982. · Zbl 0584.20036
[6] J. Cheeger, ”Finiteness theorem of Riemannian manifolds,” Amer. J. Math., 92 (1970), 61–74. · Zbl 0194.52902
[7] A. Dranishnikov, ”Infinite family of manifolds with bounded total curvature,” Proc. Amer. Math. Soc., 128:1 (2000), 255–260. · Zbl 0937.53023
[8] A. N. Dranishnikov, ”Macroscopic dimension and essential manifolds,” Proc. Steklov Inst. Math., 273 (2011), 35–47. · Zbl 1230.54027
[9] A. Dranishnikov, ”On macroscopic dimension of rationally essential manifolds,” Geometry and Topology (to appear); http://arxiv.org/abs/1005.0424 . · Zbl 1220.53057
[10] M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Birkhäuser, Boston, MA, 1999.
[11] M. Gromov, ”Positive curvature, macroscopic dimension, spectral gaps and higher signatures,” in: Functional Analysis on the Eve of the 21st Century. Vol II, Birkhäuser, Boston, MA, 1996, 1–213. · Zbl 0945.53022
[12] M. Gromov and H. B. Lawson, Jr., ”The classification of simply connected manifolds of positive scalar curvature,” Ann. of Math., 111 (1980), 209–230. · Zbl 0445.53025
[13] M. Gromov and H. B. Lawson, Jr., ”Positive scalar curvature and the Dirac operator on complete Riemannian manifolds,” Publ. Math. I.H.E.S, 58 (1983), 83–196. · Zbl 0538.53047
[14] N. Hitchin, ”Harmonic spinors,” Adv. Math., 14 (1974), 1–55. · Zbl 0284.58016
[15] A. Lichnerowicz, ”Spineurs harmoniques,” C. R. Acad. Sci. Paris, Ser. A-B, 257 (1963), 7–9. · Zbl 0136.18401
[16] J. Rosenberg, ”C*-algebras, positive scalar curvature, and the Novikov conjecture, III,” Topology, 25:3(1986), 319–336. · Zbl 0605.53020
[17] J. Rosenberg and S. Stolz, ”Metrics of positive scalar curvature and connections with surgery,” in: Surveys on Surgery Theory, vol. 2, Ann. Math. Stud., vol. 149, Princeton Univ. Press, Princeton. NJ, 2001, 353–386. · Zbl 0971.57003
[18] Yu. Rudyak, On Thom Spectra, Orientability, and Cobordism, Springer-Verlag, Berlin, 1998.
[19] T. Schick, ”Counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture,” Topology, 37:6 (1998), 1165–1168. · Zbl 0976.53052
[20] S. Stolz, ”Simply connected manifolds of positive scalar curvature,” Ann. of Math., 136:3 (1992), 511–540. · Zbl 0784.53029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.