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$$L^2$$-invariants: Theory and applications to geometry and $$K$$-theory. (English) Zbl 1009.55001
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 44. Berlin: Springer. xv, 595 p. EUR 119.00/net; sFr. 192.50; £83.50; \$ 129.00 (2002).
$$L^2$$-methods were introduced into topology by Atiyah in the 1970’s, from an analytic perspective (elliptic operators on infinite covering spaces of Riemannian manifolds). He defined $$L^2$$-Betti numbers by means of the von Neumann dimension associated to the von Neumann algebra $${\mathcal N}(\pi)$$ of the covering group $$\pi$$. Subsequently, two further types of $$L^2$$-invariants were introduced: the Novikov-Shubin invariants, which measure the asymptotic behaviour of the spectrum of the Laplace operator, and the $$L^2$$-torsion, which is an analogue of the Reidemeister torsion. Although these invariants were all originally defined in terms of elliptic operators on the $$L^2$$-completions of spaces of differential forms on the covering space, the analysis has been stripped away and the method given a more algebraic appearance, notably by Lück and Farber.
The von Neumann algebra $${\mathcal N}(\pi)$$ is the algebra of bounded (left) $$\mathbb{C}[\pi]$$-linear operators on $$\ell^2(\pi)$$, the Hilbert space completion of $$\mathbb{C}[\pi]$$. The von Neumann trace gives rise to a dimension function on finitely generated projective $${\mathcal N}(\pi)$$-modules, with values in $$[0,\infty)$$, and which extends to a $$[0,\infty]$$-valued function on arbitrary $${\mathcal N}(\pi)$$-modules. The $$L^2$$-cohomology modules of a cell-complex $$X$$ with finite skeleta and $$\pi_1(X)=\pi$$ are determined by the complex of cellular cocycles with square summable values on the universal covering of $$X$$, and the $$L^2$$-Betti numbers of $$X$$ are the dimensions of these modules. They are in general non-negative real numbers, rather than integers, but their alternating sum is the Euler characteristic $$\chi(X)$$, if $$X$$ is finitely dominated, and they satisfy Poincaré duality, if $$X$$ is a Poincaré duality complex. In addition, they are multiplicative in finite covers, and in degrees 0 and 1 depend only on the fundamental group. In Farber’s version of the theory the Novikov-Shubin invariants are invariants of the “torsion” of the $$L^2$$-cohomology modules. (The $$L^2$$-Betti numbers are invariants of the reduced $$L^2$$-cohomology, obtained by factoring out the torsion part). The $$L^2$$-torsion is defined when all the $$L^2$$-Betti numbers are 0. (We shall not attempt detailed definitions here).
The above-mentioned properties of the $$L^2$$-Betti numbers all hold for formal reasons, and the proofs are straightforward. The hard work lies in determing the nature of these invariants, in particular whether they are rational, integral or 0. The Atiyah conjecture is that the $$L^2$$-Betti numbers of a finite complex are rational, and are integral if the fundamental group is torsion-free. This is known to imply that the complex group rings of torsion-free groups have no zero-divisors. It is similarly conjectured that the Novikov-Shubin invariants of a finite complex are rational numbers (when finite). The known properties of Riemannian manifolds of constant negative curvature suggest the Singer conjecture, that the $$L^2$$-Betti numbers of an aspherical closed manifold should be nonzero only in the middle dimension. In particular, if the dimension is odd the $$L^2$$-Betti numbers should all be 0. There is then a related conjecture for the sign of the $$L^2$$-torsion of such a manifold. Vanishing criteria for some or all of the $$L^2$$-Betti numbers have surprisingly strong consequences, particularly in combinatorial group theory and low-dimensional topology.
The present book is the first substantial monograph on this topic. There are 17 numbered chapters: §0. Introduction; §1. $$L^2$$-Betti numbers; §2. Novikov-Shubin invariants; §3. $$L^2$$-torsion; §4. $$L^2$$-Invariants of 3-manifolds; §5. $$L^2$$-Invariants of symmetric spaces; §6. $$L^2$$-Invariants for general spaces with group action; §7. Application to groups; §8. The algebra of affiliated operators; §9. Middle $$K$$- and $$L$$-theory of von Neumann algebras; §10. The Atiyah conjecture; §11. The Singer conjecture; §12. The zero-in-the-spectrum conjecture; §13. The approximation and determinant conjectures; §14. $$L^2$$-Invariants and the simplicial volume; §15. Survey on other topics related to $$L^2$$-invariants; and §16. Solutions to the exercises; and a list of over 500 references.
This is an impressive account of much of what is presently known about these invariants, focused on the major applications and a number of key conjectures, with strong implications for other areas of mathematics. It combines features of a text and a reference work; to a considerable degree the chapters can be read independently, and there are numerous nontrivial exercises, with nearly 50 pages of detailed hints at the end. As the author states, it would require another book to treat in detail the topics outlined in §15. In particular, the applications to the classical knot concordance group (due to Cochran, Orr and Teichner) are probably still evolving too rapidly to be incorporated in a reference work. However, this reviewer would have liked to have seen a more substantial discussion of Farber’s approach, here confined to one paragraph in §6.8. (It should be noted also that operator algebra methods have had significant impact on other areas of manifold topology and $$K$$-theory, notably in connection with the Baum-Connes conjecture and the Novikov conjecture, but these applications involve $$C^*$$-algebras rather than von Neumann algebras).

##### MSC:
 55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology 19K99 $$K$$-theory and operator algebras 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 58-02 Research exposition (monographs, survey articles) pertaining to global analysis 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis 19-02 Research exposition (monographs, survey articles) pertaining to $$K$$-theory 55N25 Homology with local coefficients, equivariant cohomology 58J99 Partial differential equations on manifolds; differential operators