论文标题
$ l^2 $ - 无限几何的末端的元素和准等法
$L^2$-cohomology and quasi-isometries on the ends of unbounded geometry
论文作者
论文摘要
在本文中，我们研究了最低和最大的$ l^{2} $ - 定向，可能不完整的Riemannian歧管的共同体。我们的重点将放在减少和未还原的$ l^{2} $ - 共同体学组。特别是，我们将证明这些群体在无限端的均匀同型等效性下是不变的。均匀的图是一个均匀的连续图，因此子集的前图的直径是根据子集本身的直径界定的。如果$ x = m \ x = m \ x = m \ cup e_x $，其中$ x = m \ cup e_x $，其中$ f $ f $是\ textIt {quasi-is-iSometrictrict}如果$ x = m \ cup x = m \ cup e_x $，其中$ m $是$ e_x $ $ x $ $ x $的$ e_x $ for $ x $ f of $ x $ for $ x $ for $ x $ for $ x $的内部准染色。最后显示了一些后果：主要的后果是$ l^2 $的映射锥的定义，以及$ l^2 $签名的不变性。
In this paper we study the minimal and maximal $L^{2}$-cohomology of oriented, possibly not complete, Riemannian manifolds. Our focus will be on both the reduced and the unreduced $L^{2}$-cohomology groups. In particular we will prove that these groups are invariant under uniform homotopy equivalence quasi-isometric on the unbounded ends. A uniform map is a uniformly continuous map such that the diameter of the preimage of a subset is bounded in terms of the diameter of the subset itself. A map $f$ between two Riemannian manifolds $(X,g)$ and $(Y,h)$ is \textit{quasi-isometric on the unbounded ends} if $X = M \cup E_X$ where $M$ is the interior of a manifold of bounded geometry with boundary, $E_X$ is an open of $X$ and the restriction of $f$ to $E_X$ is a quasi-isometry. Finally some consequences are shown: the main ones are definition of a mapping cone for $L^2$-cohomology and the invariance of the $L^2$-signature.
