$L^2$-determinant class and approximation of $L^2$-Betti numbers
$L^2$-determinant class and approximation of $L^2$-Betti numbers
复制标题
$L^2$-行列式类和 $L^2$-Betti 数的近似值
DOI：
10.1090/s0002-9947-01-02699-x
复制
发表时间：
1998
影响因子：
1.3
通讯作者：
T. Schick
中科院分区：
文献类型：
--
作者：
T. Schick
L2-上同调中的一个猜想是：每个有限CW复形X都是L2-行列式类.在本文中，我们证明了这一点，只要基本群属于一个大的群类G，包含例如所有剩余有限群的扩张，所有剩余有限群与顺从群，所有剩余顺从群和这些群的自由积。此外，如果X是L2-非循环的，我们还证明了L2-行列式是同伦不变量.即使在已知的情况下，我们的同伦不变性的证明也比以前的证明要简短得多。在适当的条件下，给出了L2-Betti数的新的逼近公式.勘误表补充，纠正一些未经证明的声明“顺从扩张”：在整个，顺从扩张应该是正常的子群的扩展。
A standing conjecture in L 2 -cohomology is that every finite CW complex X is of L 2 -determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L 2 -acyclic, we also prove that the L 2 -determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L 2 -Betti numbers. Errata are added, rectifying some unproved statements about “amenable extension”: throughout, amenable extensions should be extensions with normal subgroups.