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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
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