"Non UBC"@en .
"DSpace"@en .
"Dranishnikov, Alexander"@en .
"2014-08-06T23:53:55Z"@en .
"2013-08-05"@en .
"In his book dedicated to Gelfand\u00E2\u0080\u0099s 80th anniversary [G] Gromov in- troduced the notion of macroscopic dimension and proposed a conjecture: The macroscopic dimension dimmcM \u00CC\u0083 of the universal cover M \u00CC\u0083 of a closed n-manifold with positive scalar curvature is at most n \u00E2\u0088\u0092 2. We proved this conjecture in [BD] for spin manifolds whose fundamental group sat- isfies the Analytic Novikov Conjecture and the following K-theoretic condition (the Rosenberg-Stolz condition [RS]): ko\u00E2\u0088\u0097(\u00CF\u0080) \u00E2\u0088\u0092\u00E2\u0086\u0092 KO\u00E2\u0088\u0097(\u00CF\u0080) is a monomorphism. In this presentation we will discuss the inequality dimmcM \u00CC\u0083 < n for for closed positive scalar curvature n-manifolds M. In particular, we prove it for manifolds whose fundamental group satisfies the Analytic Novikov Conjecture and the weaker K-theoretic condition: kolf (E\u00CF\u0080) \u00E2\u0088\u0092\u00E2\u0086\u0092 KOlf (E\u00CF\u0080) is a monomorphism. This allows to to prove the Gromov Conjecture for manifolds with the fundamental groups sat- isfying the Novikov conjecture which are duality groups. The inequality dimmcM \u00CC\u0083 < n is related to Gong-Yu\u00E2\u0080\u0099s concept of a macroscopically large manifold as well as to Gromov\u00E2\u0080\u0099s notion of inessential manifolds. We show that this inequality means exactly that M \u00CC\u0083 is macroscopically large integrally. The large obstacle on the way to the Gromov conjecture is the difference between rational and integral versions of these concepts. The\\r\\n2\\r\\nrational inessentiality means that f\u00E2\u0088\u0097([M]) = 0 in Hn(B\u00CF\u0080;Q). In the case of a spin manifold with positive scalar curvature the rational inessential- ity follows from Rosenberg\u00E2\u0080\u0099s theorem [R] and the K-homology Chern character. In [G] Gromov conjectured that the condition dimmcM \u00CC\u0083 < n implies the rational inessentiality. It turns out that this his conjecture is closely related to the question of amenability of the fundamental group [Dr2] and generally has a counterexample [Dr3].\\r\\n\\r\\nReferences:\\r\\n[BD] D. Bolotov, A. Dranishnikov, On Gromov\u00E2\u0080\u0099s scalar curvature con- jecture, Proc. of AMS, 138 no. 4 (2010), 1517-1524\\r\\n[Dr2] A. Dranishnikov, Macroscopic dimension and essential manifolds, Proceedings of Steklov Math. Institute, 273 (2011), 41-53.\\r\\n[Dr3] A. Dranishnikov, On macroscopic dimension of rationally essential mani- folds. Geometry and Topology. 15 (2011), no. 2, 1107-1124.\\r\\n[G] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signatures, Functional analysis on the eve of the 21st century. Vol II, Birhauser, Boston, MA, 1996.\\r\\n[RS] J. Rosenberg, S. Stolz, Metrics of positive scalar curvature and connections with surgery, in: Surveys on Surgery Theory (vol. 2), S. Cappell, A. Ranicki, J. Rosenberg (eds.), Annals of Mathematical Stud- ies 149 (2001), Princeton University Press.\\r\\n[R] J. Rosenberg C\u00E2\u0088\u0097-algebras, positive scalar curvature, and the Novikov con- jecture, III , Topology 25 (1986), 319 - 336\\r\\n"@en .
"https://circle.library.ubc.ca/rest/handle/2429/49422?expand=metadata"@en .
"41 minutes"@en .
"video/mp4"@en .
""@en .
"Author affiliation: University of Florida"@en .
"10.14288/1.0043476"@en .
"eng"@en .
"Unreviewed"@en .
"Vancouver : University of British Columbia Library"@en .
"Banff International Research Station for Mathematical Innovation and Discovery"@en .
"Attribution-NonCommercial-NoDerivs 2.5 Canada"@en .
"http://creativecommons.org/licenses/by-nc-nd/2.5/ca/"@en .
"Faculty"@en .
"BIRS Workshop Lecture Videos (Banff, Alta)"@en .
"Mathematics"@en .
"Differential geometry"@en .
"Manifolds and cell complexes"@en .
"Scalar curvature, macroscopic dimension and inessential manifolds"@en .
"Moving Image"@en .
"http://hdl.handle.net/2429/49422"@en .