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Scalar curvature, macroscopic dimension and inessential manifolds Dranishnikov, Alexander
Description
In his book dedicated to Gelfand’s 80th anniversary [G] Gromov in troduced the notion of macroscopic dimension and proposed a conjecture: The macroscopic dimension dimmcM ̃ of the universal cover M ̃ of a closed nmanifold with positive scalar curvature is at most n − 2. We proved this conjecture in [BD] for spin manifolds whose fundamental group sat isfies the Analytic Novikov Conjecture and the following Ktheoretic condition (the RosenbergStolz condition [RS]): ko∗(π) −→ KO∗(π) is a monomorphism. In this presentation we will discuss the inequality dimmcM ̃ < n for for closed positive scalar curvature nmanifolds M. In particular, we prove it for manifolds whose fundamental group satisfies the Analytic Novikov Conjecture and the weaker Ktheoretic condition: kolf (Eπ) −→ KOlf (Eπ) 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 ̃ < n is related to GongYu’s concept of a macroscopically large manifold as well as to Gromov’s notion of inessential manifolds. We show that this inequality means exactly that M ̃ 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∗([M]) = 0 in Hn(Bπ;Q). In the case of a spin manifold with positive scalar curvature the rational inessential ity follows from Rosenberg’s theorem [R] and the Khomology Chern character. In [G] Gromov conjectured that the condition dimmcM ̃ < 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’s scalar curvature con jecture, Proc. of AMS, 138 no. 4 (2010), 15171524\\r\\n[Dr2] A. Dranishnikov, Macroscopic dimension and essential manifolds, Proceedings of Steklov Math. Institute, 273 (2011), 4153.\\r\\n[Dr3] A. Dranishnikov, On macroscopic dimension of rationally essential mani folds. Geometry and Topology. 15 (2011), no. 2, 11071124.\\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∗algebras, positive scalar curvature, and the Novikov con jecture, III , Topology 25 (1986), 319  336\\r\\n
Item Metadata
Title 
Scalar curvature, macroscopic dimension and inessential manifolds

Creator  
Publisher 
Banff International Research Station for Mathematical Innovation and Discovery

Date Issued 
20130805

Description 
In his book dedicated to Gelfand’s 80th anniversary [G] Gromov in troduced the notion of macroscopic dimension and proposed a conjecture: The macroscopic dimension dimmcM ̃ of the universal cover M ̃ of a closed nmanifold with positive scalar curvature is at most n − 2. We proved this conjecture in [BD] for spin manifolds whose fundamental group sat isfies the Analytic Novikov Conjecture and the following Ktheoretic condition (the RosenbergStolz condition [RS]): ko∗(π) −→ KO∗(π) is a monomorphism. In this presentation we will discuss the inequality dimmcM ̃ < n for for closed positive scalar curvature nmanifolds M. In particular, we prove it for manifolds whose fundamental group satisfies the Analytic Novikov Conjecture and the weaker Ktheoretic condition: kolf (Eπ) −→ KOlf (Eπ) 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 ̃ < n is related to GongYu’s concept of a macroscopically large manifold as well as to Gromov’s notion of inessential manifolds. We show that this inequality means exactly that M ̃ 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∗([M]) = 0 in Hn(Bπ;Q). In the case of a spin manifold with positive scalar curvature the rational inessential ity follows from Rosenberg’s theorem [R] and the Khomology Chern character. In [G] Gromov conjectured that the condition dimmcM ̃ < 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’s scalar curvature con jecture, Proc. of AMS, 138 no. 4 (2010), 15171524\\r\\n[Dr2] A. Dranishnikov, Macroscopic dimension and essential manifolds, Proceedings of Steklov Math. Institute, 273 (2011), 4153.\\r\\n[Dr3] A. Dranishnikov, On macroscopic dimension of rationally essential mani folds. Geometry and Topology. 15 (2011), no. 2, 11071124.\\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∗algebras, positive scalar curvature, and the Novikov con jecture, III , Topology 25 (1986), 319  336\\r\\n

Extent 
41 minutes

Subject  
Type  
File Format 
video/mp4

Language 
eng

Notes 
Author affiliation: University of Florida

Series  
Date Available 
20140806

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivs 2.5 Canada

DOI 
10.14288/1.0043476

URI  
Affiliation  
Peer Review Status 
Unreviewed

Scholarly Level 
Faculty

Rights URI  
Aggregated Source Repository 
DSpace

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Item Citations and Data
Rights
AttributionNonCommercialNoDerivs 2.5 Canada