Abstract
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 \(\tilde M\) is strictly less than n: dim MC \(\tilde M < n\). As a corollary, we obtain the following partial result towards Gromov’s conjecture
The inequality dim MC \(\tilde M < n\) holds for the universal cover \(\tilde M\) of a closed spin n-manifold M with a positive scalar curvature metric if the fundamental group π 1 (M) is a duality group satisfying the analytic Novikov conjecture.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. I. Arnold, “Dynamics of complexity of intersections,” Bol. Soc. Brasil. Mat. (N.S.), 21:1 (1990), 1–10.
D. Bolotov, “Macroscopic dimension of 3-manifolds,” Math. Phys. Anal. Geom., 6 (2003), 291–299.
D. Bolotov and A. Dranishnikov, “On Gromov’s scalar curvature conjecture,” Proc. Amer. Math. Soc., 138:4 (2010), 1517–1524.
S. Brendle and R. Schoen, “Sphere theorems in geometry,” in: Surv. Differential Geometry, vol. XIII, Intern. Press, Somerville, MA, 2009, 49–84.
K. Brown, Cohomology of Groups, Springer-Verlag, New York-Berlin, 1982.
J. Cheeger, “Finiteness theorem of Riemannian manifolds,” Amer. J. Math., 92 (1970), 61–74.
A. Dranishnikov, “Infinite family of manifolds with bounded total curvature,” Proc. Amer. Math. Soc., 128:1 (2000), 255–260.
A. N. Dranishnikov, “Macroscopic dimension and essential manifolds,” Proc. Steklov Inst. Math., 273 (2011), 35–47.
A. Dranishnikov, “On macroscopic dimension of rationally essential manifolds,” Geometry and Topology (to appear); http://arxiv.org/abs/1005.0424.
M. Gromov, Metric Structures for Riemannian and non-Riemannian Spaces, Birkhäuser, Boston, MA, 1999.
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.
M. Gromov and H. B. Lawson, Jr., “The classification of simply connected manifolds of positive scalar curvature,” Ann. of Math., 111 (1980), 209–230.
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.
N. Hitchin, “Harmonic spinors,” Adv. Math., 14 (1974), 1–55.
A. Lichnerowicz, “Spineurs harmoniques,” C. R. Acad. Sci. Paris, Ser. A-B, 257 (1963), 7–9.
J. Rosenberg, “C*-algebras, positive scalar curvature, and the Novikov conjecture, III,” Topology, 25:3(1986), 319–336.
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.
Yu. Rudyak, On Thom Spectra, Orientability, and Cobordism, Springer-Verlag, Berlin, 1998.
T. Schick, “Counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture,” Topology, 37:6 (1998), 1165–1168.
S. Stolz, “Simply connected manifolds of positive scalar curvature,” Ann. of Math., 136:3 (1992), 511–540.
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 45, No. 3, pp. 34–40, 2011
Original Russian Text Copyright © by A. N. Dranishnikov
To the memory of V. I. Arnold
Supported by NSF grant DMS-0904278.
Rights and permissions
About this article
Cite this article
Dranishnikov, A.N. On macroscopic dimension of rationally inessential manifolds. Funct Anal Its Appl 45, 187–191 (2011). https://doi.org/10.1007/s10688-011-0022-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10688-011-0022-9