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$$L^ 2$$-invariants of manifolds and groups. ($$L^ 2$$-Invarianten von Mannigfaltigkeiten und Gruppen.) (German) Zbl 0881.57021
\loadmsbm This is a short overview about $$L^2$$-invariants of manifolds. If $$X$$ is a CW-complex of finite type and if $$\overline X\to X$$ is a regular covering with covering group $$\Gamma$$, then one can define the $$p$$th $$L^2$$-homology of $$\overline X$$, which is a Hilbert space with an isometric $$\Gamma$$-action on it. Its von-Neumann-dimension is called the $$p$$th $$L^2$$-Betti number $$b^{(2)}_p(\overline X)$$ of $$\overline X$$. The $$L^2$$-Betti numbers $$b_p^{(2)}(\overline X)$$ share many of the properties of the Betti numbers $$b_p(X)$$.
If $$X=M$$ is a compact orientable Riemannian manifold with covering $$\overline M$$, then $$b_p^{(2)}(\overline M)$$ can be defined using Hodge theory: Let $$e^{-t\Delta_p}$$ be the heat kernel on the $$p$$-forms on $$\overline M$$ and let $$\mathcal F$$ be a fundamental domain for the $$\Gamma$$-action on $$\overline M$$, then $b^{(2)}_p(\overline M) =\lim_{t\to\infty}\int_{\mathcal F} \text{tr}_{\mathbb{R}}\bigl(e^{-t\Delta_p}(\overline x,\overline x)\bigr) d\overline x .$
The author mentions some conjectures concerning $$L^2$$-Betti numbers, among them the Atiyah conjecture, which says that the $$L^2$$-Betti numbers of the universal cover $$\widetilde M$$ of a closed differentiable manifold $$M$$ are rational, and integer, if the fundamental group of $$M$$ has no torsion. The Singer conjecture says that $$b^{(2)}_p(\widetilde M)=0$$ for $$p\neq n/2$$ if $$\widetilde M$$ is the universal cover of an $$n$$-dimensional compact Riemannian manifold $$M$$ of nonpositive sectional curvature. The Hopf conjecture says that if the sectional curvature of $$M$$ is strictly negative and $$n$$ is even, then additionally $$b^{(2)}_{n/2}(\widetilde M)>0$$. All of these conjectures have been proven for some special cases.
The author presents some more results on $$L^2$$-Betti numbers and sketches the definition and some properties of the $$L^2$$-analogues of Reidemeister and Ray-Singer torsion.
##### MSC:
 57Q99 PL-topology 57R57 Applications of global analysis to structures on manifolds 57M10 Covering spaces and low-dimensional topology