A ``stable'' version of the Gromov-Lawson conjecture
A ``stable'' version of the Gromov-Lawson conjecture
复制标题
格罗莫夫-劳森猜想的“稳定”版本
DOI：
10.1090/conm/181
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发表时间：
1994
期刊：
影响因子：
--
通讯作者：
S. Stolz
中科院分区：
文献类型：
--
作者：
Jonathan Rosenberg;S. Stolz
我们讨论了Gromov和Lawson的一个猜想，后来由Rosenberg修改，关于度量的存在性的正标量曲率。它说一个维数为$n\ge 5 $的闭自旋流形$M $有这样一个度量当且仅当$KO_n（C ^*（\pi_1（M）$中一个合适的"狄拉克"算子的指数，即$M $基本群的群$C ^*$-代数的真实的$K $-理论为零。已知指数为零是$M $上存在正数量曲率度量的必要条件，但这仅当$\pi_1（M）$是平凡群、$\Bbb Z/2 $、奇阶循环群或一个相当小的无挠群时才是充分条件。我们注意到群$KO_n（C ^*（\pi））$在$n $中是周期的，周期为$8 $，而在原始几何问题中没有明显的周期性。这导致我们引入Gromov-Lawson猜想的一个"稳定"版本，它使得较弱的陈述，即$M $与足够多的"Bott流形"$B $的乘积具有正的标量曲率度量当且仅当$M $上的Dirac算子的指数为零。(Here$B $是一个单连通的$8 $-流形，它表示$KO_8（pt）$中的周期元。证明了Gromov-Lawson猜想对所有具有有限基本群的自旋流形和许多具有无限基本群的自旋流形的稳定性。
We discuss a conjecture of Gromov and Lawson, later modified by Rosenberg, concerning the existence of metrics of positive scalar curvature. It says that a closed spin manifold $M$ of dimension $n\ge 5$ has such a metric if and only if the index of a suitable ``Dirac" operator in $KO_n(C^* (\pi_1(M)))$, the real $K$-theory of the group $C^*$-algebra of the fundamental group of $M$, vanishes. It is known that the vanishing of the index is necessary for existence of a positive scalar curvature metric on $M$, but this is known to be a sufficient condition only if $\pi_1(M)$ is the trivial group, $\Bbb Z/2$, an odd order cyclic group, or one of a fairly small class of torsion-free groups. \par We note that the groups $KO_n(C^*(\pi))$ are periodic in $n$ with period $8$, whereas there is no obvious periodicity in the original geometric problem. This leads us to introduce a ``stable'' version of the Gromov-Lawson conjecture, which makes the weaker statement that the product of $M$ with enough copies of the ``Bott manifold" $B$ has a positive scalar curvature metric if and only if the index of the Dirac operator on $M$ vanishes. (Here $B$ is a simply connected $8$-manifold which represents the periodicity element in $KO_8(pt)$.) We prove the stable Gromov-Lawson conjecture for all spin manifolds with finite fundamental group and for many spin manifolds with infinite fundamental group.