No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds

A metric space $X$ is called uniformly acyclic if there there exists an {\it acyclicty control function} $R=R(r)=R_X(r)\geq r$, $0\leq r <\infty$, such that the homology inclusion homomorphisms between the balls around all points $x\in X$, $H_i(B_x(r))\to H_i(B_x(R))$ vanish for all $i=1,2,\ldots$. We show that if a complete orientable $m$-dimensional manifold $\tilde X$ of dimension $m\leq 5$ admits a proper (infinity goes to infinity) distance decreasing map to a complete $m$-dimensional uniformly acyclic manifold, then the scalar curvature of $\tilde X$ can't be uniformly positive, $\inf _{x\in \tilde X}Sc(X,x) \leq 0.$ Since the universal coverings $\tilde X$ of compact aspherical manifolds $X$ are {\it uniformly acyclic}, (in fact, {\it uniformly contractible}), these $X$, admit no metrics with $Sc>0$ for $dim (X)\leq 5$. Our argument, that depends on {\it torical symmetrization} of {\it stable $μ$-bubbles}, is inspired by the recent paper by Otis Chodosh and Chao Li on non-existence of metrics with $Sc>0$ on aspherical 4-manifolds and is also influenced by the ideas of Jintian Zhu and Thomas Richard.

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