论文标题
没有在非球面5个manifolds上具有正标曲率的指标
No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds
论文作者
论文摘要
指标空间$ x $如果存在{\ it acybilty控制函数} $ r = r = r = r_x（r）= r_x（r）\ geq r $，$ 0 \ leq r <\ r <\ infty $ H_i（b_x（r））$$消失了所有$ i = 1,2，\ ldots $。我们表明，如果一个完全定位的$ m $ $二维流形$ \ tilde x $ dimension $ m \ m \ leq 5 $承认，适当的（无穷大）距离降低了地图，将图降低到一个完整的$ m $ b $尺寸均匀均匀的cycyclic cycyclic流动，然后是$ \ tilde x $ sifort $ \ tilde undirild f.tird y Infly， x} sc（x，x）\ leq 0。$$，因为通用覆盖物$ \ tilde x $ x $ $ x $是{\ it均匀地是acyclic}，（实际上，{\ it {\ it均匀签约}），这些$ x $，这些$ x $，no n n n n n no n n n n n n n nime sc> sc> 0 $ 0 $ $ $ dim $ dim dim dim dim（x）我们的论点取决于{\ IT稳定的$μ$ $ - bubbles}的{\ It torical对称性}，这是受到Otis Chodosh和Chao li的最新论文的启发。
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.
