[Paper Review] On Gromov's positive scalar curvature conjecture for virtual duality groups

Alexander Dranishnikov|arXiv (Cornell University)|Dec 10, 2013

Geometric and Algebraic Topology20 references3 citations

TL;DR

This paper proves Gromov's conjecture that the macroscopic dimension of the universal cover of a closed almost spin $n$-manifold with positive scalar curvature is at most $n-2$, provided the fundamental group is a virtual duality group satisfying the coarse Baum-Connes conjecture. The proof uses coarse cohomology, the Berstein-Schwarz class, and obstruction theory to show that the primary obstruction to deformation into the $(n-2)$-skeleton vanishes, extending the class of groups for which the conjecture holds beyond previous results.

ABSTRACT

We prove the inequality $$ \dim_{mc}\Wi M\le n-2$$ for the macroscopic dimension of the universal covers $\Wi M$ of almost spin $n$-manifolds $M$ with positive scalar curvature whose fundamental group $π_1(M)$ is a virtual duality group that satisfies the coarse Baum-Connes conjecture.

Motivation & Objective

• To extend Gromov's positive scalar curvature conjecture to a broader class of fundamental groups, including virtually nilpotent, arithmetic, and mapping class groups.
• To establish the inequality $\dim_{mc}\widetilde{M} \leq n-2$ for almost spin manifolds with virtual duality fundamental groups satisfying the coarse Baum-Connes conjecture.
• To generalize the notion of macroscopic dimension to include coarse inessentiality and relate it to rational inessentiality and group homology.
• To reduce the weak Gromov conjecture to a rationality conjecture on the Stone-Čech compactification and cohomology classes of $\beta\widetilde{M}$.

Proposed method

• Uses the macroscopic dimension $\dim_{mc}$ defined via uniformly cobounded maps to finite-dimensional simplicial complexes.
• Applies coarse cohomology and the coarse Baum-Connes conjecture to analyze the large-scale geometry of $\widetilde{M}$.
• Employs the Berstein-Schwarz class $b \in H^1(\pi, I(\pi))$ and its cup powers to construct obstructions to deformation into lower skeleta.
• Reduces the problem to the vanishing of the primary obstruction $o_{\widetilde{f}}$ via the universal property of the Berstein-Schwarz class.
• Uses the $\pi_1$-equivariant perturbation map $Pert_*$ and the coefficient homomorphism $ec^*_M$ to relate fundamental classes to cohomological obstructions.
• Relies on the rationality conjecture for sheaf cohomology on $\beta\widetilde{M}$ to conclude that the obstruction class has finite order and thus vanishes.

Experimental results

Research questions

• RQ1 Does the macroscopic dimension of the universal cover of a positive scalar curvature manifold satisfy $\dim_{mc}\widetilde{M} \leq n-2$ for virtual duality groups satisfying the coarse Baum-Connes conjecture?
• RQ2 Can the weak Gromov conjecture ($\dim_{mc}\widetilde{M} \leq n-1$) be reduced to a rationality condition on the Stone-Čech compactification of $\widetilde{M}$?
• RQ3 Is the primary obstruction to deforming the classifying map into the $(n-1)$-skeleton trivial when the fundamental class is torsion in rational homology?
• RQ4 To what extent does the Berstein-Schwarz class control the macroscopic dimension via its cup powers and coefficient homomorphisms?
• RQ5 Does the vanishing of the obstruction class $o_{\widetilde{f}}$ imply $\dim_{mc}\widetilde{M} < n$ for almost spin manifolds with such fundamental groups?

Key findings

• The inequality $\dim_{mc}\widetilde{M} \leq n-2$ holds for all almost spin $n$-manifolds with positive scalar curvature and fundamental group a virtual duality group satisfying the coarse Baum-Connes conjecture.
• The proof establishes that the primary obstruction $o_{\widetilde{f}}$ to deformation into the $(n-2)$-skeleton vanishes, implying $\dim_{mc}\widetilde{M} < n$.
• The rational inessentiality of $M$ implies that $f_*([M])$ is torsion, which forces $f^*(b^n)$ to have finite order, leading to vanishing of the obstruction in the rational setting.
• The weak Gromov conjecture is reduced to a rationality conjecture on the cohomology of $\beta\widetilde{M}$ with coefficients in $\bar{p}^*\mathcal{I}^n$, under which the obstruction class $\tilde{o}$ vanishes.
• The result extends the class of groups for which Gromov's conjecture holds to include virtually nilpotent groups, arithmetic groups, knot groups, braid groups, and $\mathrm{Out}(F_n)$, beyond products of free groups.
• The equivalence between $\dim_{mc}\widetilde{M} < n$ and the vanishing of $\widetilde{f}_*([\widetilde{M}])$ in $H^{lf}_n(E\pi;\mathbb{Z})$ is established via coarse homology and perturbation theory.