Scalar curvature, macroscopic dimension and inessential manifolds

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 n-manifold 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 K-theoretic condition (the Rosenberg-Stolz 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 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 (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 Gong-Yu’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 K-homology 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), 1517-1524\\r\\n[Dr2] A. Dranishnikov, Macroscopic dimension and ...