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On macroscopic dimension of universal coverings of closed manifolds. (English) Zbl 1310.55005

Trans. Mosc. Math. Soc. 2013, 229-244 (2013) and Tr. Mosk. Mat. O.-va 74, No. 2, 279-296 (2013).
One can make the following definition for an arbitrary metric space \(X\):
Definition: \(\dim_{mc}X\leq n\) if there is a map \(g\) of \(X\) to a simplicial complex of dimension \(\leq n\) such that for some \(b>0\), \(\mathrm{diam }g^{-1}(y)<b\) for all \(y\) in the range.
This “macroscopic” dimension was defined by M. Gromov [Prog. Math. 132, 1–213 (1996; Zbl 0945.53022)] where \(X=\widetilde M\) for \(M\) a Riemannian manifold and \(\widetilde M\) its universal covering. On the other hand, Dranishnikov has defined a different, somewhat similar, notion of macroscopic dimension denoted \(\dim_{MC}X\) (see p. 230), also for an arbitrary metric space \(X\). One may ask to what extent \(\dim_{mc}\) and \(\dim_{MC}\) are distinct.
The author proves that \(\dim_{mc}\widetilde M<\dim_{MC}\widetilde M=n\) for the universal covering \(\widetilde M\) of any closed \(n\)-manifold \(M\) in case the fundamental group \(\pi\) of \(M\) is a geometrically finite amenable duality group with cohomological dimension \(cd(\pi)>n\). This distinguishes \(\dim_{mc}\) from \(\dim_{MC}\). The techniques involve obtaining a homological characterization of \(n\)-manifolds whose universal covering has Gromov’s macroscopic dimension \(<n\).

MSC:

55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects)
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57N65 Algebraic topology of manifolds

Citations:

Zbl 0945.53022
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References:

[1] G. C. Bell and A. N. Dranishnikov, A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory, Trans. Amer. Math. Soc. 358 (2006), no. 11, 4749 – 4764. · Zbl 1117.20032
[2] Israel Berstein, On the Lusternik-Schnirelmann category of Grassmannians, Math. Proc. Cambridge Philos. Soc. 79 (1976), no. 1, 129 – 134. · Zbl 0315.55011 · doi:10.1017/S0305004100052142
[3] Dmitry V. Bolotov, Macroscopic dimension of 3-manifolds, Math. Phys. Anal. Geom. 6 (2003), no. 3, 291 – 299. · Zbl 1029.57016 · doi:10.1023/A:1024994930786
[4] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. · Zbl 0584.20036
[5] Michael Brunnbauer and Bernhard Hanke, Large and small group homology, J. Topol. 3 (2010), no. 2, 463 – 486. · Zbl 1196.53028 · doi:10.1112/jtopol/jtq014
[6] Dmitry Bolotov and Alexander Dranishnikov, On Gromov’s scalar curvature conjecture, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1517 – 1524. · Zbl 1193.53106
[7] Alexander Dranishnikov, On macroscopic dimension of rationally essential manifolds, Geom. Topol. 15 (2011), no. 2, 1107 – 1124. · Zbl 1220.53057 · doi:10.2140/gt.2011.15.1107
[8] A. N. Dranishnikov, Macroscopic dimension and essential manifolds, Tr. Mat. Inst. Steklova 273 (2011), no. Sovremennye Problemy Matematiki, 41 – 53; English transl., Proc. Steklov Inst. Math. 273 (2011), no. 1, 35 – 47. · Zbl 1230.54027 · doi:10.1134/S0081543811040043
[9] Alexander N. Dranishnikov and Yuli B. Rudyak, On the Berstein-Svarc theorem in dimension 2, Math. Proc. Cambridge Philos. Soc. 146 (2009), no. 2, 407 – 413. · Zbl 1171.55002 · doi:10.1017/S0305004108001904
[10] S. M. Gersten, Cohomological lower bounds for isoperimetric functions on groups, Topology 37 (1998), no. 5, 1031 – 1072. · Zbl 0933.20026 · doi:10.1016/S0040-9383(97)00070-0
[11] 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. · Zbl 1262.90126 · doi:10.1007/s10107-010-0354-x
[12] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1 – 295. · Zbl 0841.20039
[13] John Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, American Mathematical Society, Providence, RI, 2003. · Zbl 1042.53027
[14] Schwarz, A.S., The genus of a fiber space. AMS Transl., Vol. 55, No. 2, 1966, pp. 49-140.
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