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Summary of research projects of Wolfgang Lück(Bonn)date: February 27, 2014I have worked and contributed to the following areas:• Manifold topology• L 2 -invariants• K-theory• Transformation groups• Non-commutative geometry• Global analysisHere are some major projects which I have worked on during the last years.Some of them are finished, some are work in progress.1 Borel Conjecture and Farrell-Jones ConjectureIn the paper [7] with Bartels and Reich we prove the Farrell-Jones Conjecturefor algebraic K-theory with coefficient in additive categories for group whichare subgroups of finite products of hyperbolic groups or colimits of directedsystems of hyperbolic groups. The ingredients are controlled topology and flowson compactifications of a model of the classifying space for proper actions. It hasmany applications to conjectures by Bass, Moody and Kaplansky. The relevantpapers are [7] and [8]. The paper [6] plays an important role in the proof.The L-theory version is proved in the paper [4] joint with Bartels, where wealso deal with CAT(0)-groups. Here also the construction of a geodesic flow forCAT(0)-spaces plays a role, see [5]). This is probably one of my best results ofthe last five years and is a result of a project I have worked on for the last decade.It implies both the Novikov Conjecture and the Borel Conjecture for such groupsin high dimensions. The Borel Conjecture is the topological version of Mostowrigidity. It predicts for a group G that two closed aspherical manifolds M andN whose fundamental groups are isomorphic to G are homeomorphic and thatevery homotopy equivalence M → N is homotopic to a homeomorphism.We can also show (see [1]) that both the Bost Conjecture and the Farrell-Jones Conjecture with coefficients are true for those groups for which Higson-Lafforgue-Skandalis have disproved the Baum-Connes Conjecture with coefficients.1

4 L 2 -InvariantsA book on this topic [46] has appeared in 2002. It contains a detailed presentationsof the foundations, main theorems and main open problems aboutL 2 -invariants and includes the (at the time of writing) most recent developments.My own main contributions are besides some new material in the book [46]above the approximation theorem in [36] and the vanishing theorem for L 2 -Bettinumbers of mapping tori in [37] which both prove conjectures of Gromov, thecomputation of L 2 -invariant of compact three-manifolds in [33] and [73], andthe dimension theory for arbitrary modules over finite von Neumann algebrasin [39] and [40].Here are some recent results:• “L 2 -torsion and orbit equivalence”Gaboriau has shown that L 2 -Betti numbers are invariants under orbitequivalence of groups. Roland Sauer, Christian Wegner and I want toattack the conjecture that the vanishing of the L 2 -torsion is an invariantunder orbit equivalence if the groups have cocompact models for theclassifying space for proper actions and all their L 2 -Betti numbers vanish.Partial results can be found in the preprint [72].• The limit of F p -Betti numbers of a tower of finite covers withamenable fundamental groupsWe prove an analogue of the Approximation Theorem of L 2 -Betti numbersby Betti numbers for arbitrary coefficient fields and virtually torsionfreeamenable groups. The limit of Betti numbers is identified as the dimensionof some module over the Ore localization of the group ring. See [32].Bergeron-Lück and Sauer [11] study the asymptotic growth of Betti numbersin p-adic analytic towers of covers which is related to results of Calegariand Emerton, and will also present some results for pro-p towers thatare not necessarily p-adic analytic.• Approximating the first L 2 -Betti number of residually finite groupsIn the paper Lück-Osin [64] we show that the first L 2 -Betti number of afinitely generated residually finite group can be estimated from below byusing ordinary first Betti numbers of finite index normal subgroups. Asan application we construct a finitely generated infinite residually finitetorsion group with positive first L 2 -Betti number.Ershof and Lück [17] give examples, where the F p -approximation and theQ-approximation for the first Betti numbers of a tower of normal subgroupsof finite index with trivial intersection lead to different values for afinitely generated group. This group is not finitely presented. So the questionremains open whether we have equality for finitely presented groups.3

7 The Burnside Ring and Equivariant StableCohomotopy for Infinite GroupsIn [49] we propose several definitions for the Burnside ring of an infinite group.We introduce stable equivariant cohomotopy for proper cocompact G-CW -complexes. We formulate a version of the Segal Conjecture for infinite groups inthis setting. We have already a promising strategy for its proof and the Atiyah-Segal completion theorem for proper actions of an infinite discrete group hasalready been proved in [62, 63]. If the Segal Conjecture for infinite groupsis true, it seem to be justified to proceed along these lines further and enternew territory. We also want to establish the equivariant stable homotopy categoryfor non-compact Lie groups, having loop groups and Kac-Moody groupsin mind. There are interesting analogies between the hypothetical equivariantK-homology of the classifying space for proper group actions for certain loopgroups and the hypothetical topological K-theory of the hypothetical group C ∗ -algebra coming from computations of the first group in terms of representationtheory of the loop group. This suggest that there might be a version of theBaum-Connes Conjecture in this setting.Together with Schwede we are working on a model structure on the equivarianthomotopy category for infinite groups.8 Computational aspects of the Farrell-Jones ConjectureVia the conjectures due to Baum-Connes and Farrell-Jones one can try to computethe algebraic K and L-theory of group rings and the topological K-theoryof reduced group C ∗ -algebras by investigating the source of the assembly map.Rational the equivariant Chern characters give some general information, integralcomputations seem only to be possible in special cases. We have workedout details for hyperbolic groups and certain virtually Z n -groups, see [14, 70].In a project joined with Jim Davis we want to apply these to classify manifoldswhich occur as total spaces of bundles with a torus a fiber and a lens space withcyclic group of prime order as base.Consider the semi-direct product Z n ⋊ ρ Z/m. A conjecture of Adem-Ge-Pan-Petrosyan predicts that the associated Lyndon-Hochschild-Serre spectralsequence collapses. We prove this conjecture provided that the Z/m-action onZ n is free outside the origin. Langer-Lück[29] disprove the conjecture in general,namely, we give an example with n = 6 and m = 4, where the second differentialdoes not vanish. The topological K-theory of the associated group C ∗ -algebrais computed in [30] and plays a role the paper of Li-Lück [31], where the K-theory for ring C ∗ -algebras in the case of higher roots of unity and therebycompletely determine the K-theory for ring C ∗ -algebras attached to rings ofintegers in arbitrary number fields. By Kirchberg’s results this leads also to afull classification of these C ∗ -algebras.5

9 Classing spaces for familiesWe investigate the notion of classifying spaces for families which is due to tomDieck and has gotten new attention in connection with the Baum-Connes Conjectureand the Farrell-Jones Conjecture. The following papers deal with theversions, where the family is finite or virtually cyclic [41, 51, 61, 77]. The paper[55] deals with the classifying space for the virtually cyclic subgroups of agroup which acts properly cocompactly and isometrically on a CAT(0)-space.10 Surgery theoryThis is one of my early big projects. The relevant papers are [27, 28, 47, 59, 60,65, 66]. At that time the papers with Madsen [59, 60] had some impact but theyaddress the field of equivariant surgery which is not so active anymore. Therewe prove a splitting result for the equivariant L-theory computing it in termsof classical L-groups using transfers. If the group under consideration has oddorder, this splitting is just a direct sum decomposition since in this situation allthe transfer maps are trivial.Crowley, Macko and I are writing a book on surgery theory.11 Equivariant Topology of Configuration SpacesBlagojević, Lück, Wolfgang and Ziegler [12] study the Fadell–Husseini indexof the configuration space F (R d , n) with respect to different subgroups of thesymmetric group Sym n . For p prime and k ≥ 1, we completely determineIndex Z/p (F (R d , p); F p ). In this process we obtain results of independent interest,including: (1) an extended equivariant Goresky–MacPherson formula, (2) acomplete description of the top homology of the partition lattice Π p as an F p [Z p ]-module, and (3) a generalized Dold theorem for elementary abelian groups.The results on the Fadell–Husseini index yield a new proof of the Nandakumar& Ramana Rao conjecture for a prime. Moreover, for n = p k a prime power,we compute the Lusternik–Schnirelmann category cat(F (R d , n)/ Sym n ) = (d −1)(n−1), and for spheres obtain the bounds (d−1)(n−1) ≤ cat(F (S d , n)/ Sym n ) ≤(d − 1)(n − 1) + 1.Blagojević, Lück, Wolfgang and Ziegler [13] study k-regular maps R d →R N , i.e., continuous maps which send any k pairwise disjoint points to k linearindependent vectors. We show that N has to exceed a certain expression interms of d and k if such a map exists.12 Fibering manifoldsThis is a joint work in progress with Tom Farrell and Wolfgang Steimle. Thequestion is when a map between closed manifolds is homotopy equivalent to theprojection of a fiber bundle of closed manifolds. This problem was solved by6

Farrell in the case where the target is S 1 . We have figured out the generalizationof the torsion obstructions of Farrell to the general case. However, it is obviousthat in general the vanishing of this invariants is necessary but not sufficientunless the target is S 1 . There exists already the paper [18] about this project.The papers [74] and [75] are relevant for this project.13 Some smaller projects• “Crossed products of irrational rotation algebras by finite groups”This is a joint project with with Siegfried Echterhoff, Chris Phillips andSam Walters. In [16] we show that the crossed products A θ ⋊ α F and thefixed-point algebras A F θfor the action of any finite subgroup F ⊆ SL 2(Z)(which are isomorphic to one of the cyclic groups Z 2 , Z 3 , Z 4 , Z 6 ) on theirrational rotational algebra A θ via the restriction of the canonical actionof SL 2 (Z) on A θ are AF algebras. The same is shown for the flip actionof Z 2 on any simple d-dimensional torus A θ . On the way, we shall prove anumber of general results which should have useful applications in othersituations.For this paper computations about the topological K-theory of the reducedgroup C ∗ -algebra of certain finite extension of the three-dimensionalHeisenberg group were crucial (see [50]).• “Topological rigidity of non-aspherical manifolds”We investigate in [26] for which non-aspherical manifolds the Borel-Conjectureor certain variations of it are still true. A closed connected topologicalmanifold M is called topologically rigid if any homotopy equivalenceN → M of closed topological manifolds is homotopic to a homeomorphism.The Borel Conjecture for a closed aspherical manifold is equivalent to thestatement that M is a topologically rigid. We investigate when a possiblynon-aspherical manifold is topologically rigid. We show for a closedconnected orientable surface K different from S 2 , that K × S d is topologicallyrigid for d ≥ 3. For product of spheres the answer is related tothe Arf-invariant-one-problem. A closed 3-manifold is topologically rigidif and only if its fundamental group is torsionfree. The connected sumof two topologically rigid closed manifolds with torsionfree fundamentalgroup and dimension ≥ 5 is again topologically rigid.• “Finiteness obstructions and Euler characteristics of categories”This is a joint project with Roman Sauer and Tom Fiore. We want togeneralize the definition of the Euler characteristic due to Leinster and theMöbius inversions for finite categories to EI-categories. We have already avery general K-theoretic Möbius inversion which seems to encompass thestandard notions of Möbius inversion in combinatorics. See [21] and [20].7

• “A cocompletion theorem in K-theory for proper actions of adiscrete group”This refers to a paper joint with Michael Joachim [22] Let G be a discretegroup. We give methods to compute for a generalized (co-)homologytheory its values on the Borel construction EG × G X of a proper G-CW -complex X satisfying certain finiteness conditions. In particular wegive formulas computing the topological K-(co)homology K ∗ (BG) andK ∗ (BG) up to finite abelian torsion groups. They apply for instance toarithmetic groups, word hyperbolic groups, mapping class groups and discretecocompact subgroups of almost connected Lie groups. For finitegroups G these formulas are sharp. The main new tools we use for theK-theory calculation are a Cocompletion Theorem and Equivariant UniversalCoefficient Theorems which are of independent interest. In the casewhere G is a finite group these theorems reduce to well-known results ofGreenlees and Bökstedt. The cocompletion Theorem is obtained by dualizingthe equivariant version of the Atiyah-Segal completion theorem forproper actions of an infinite discrete group G in [62, 63].• “Waldhausen Additivity: Classical and Quasicategorical”Fiore and Lück [19] give a short proof of classical Waldhausen Additivity,and then prove Waldhausen Additivity for an ∞-version of WaldhausenK-theory. Namely, we prove that Waldhausen K-theory sends a splitexactsequence of Waldhausen quasicategories A → E → () to a stableequivalence of spectra K(E) → K(A) ∨ K(()) under a few mild hypotheses.For example, each cofiber sequence in A of the form A 0 → A 1 → ∗is required to have the first map an equivalence. Model structures, presentability,and stability are not needed. In an effort to make the articleself-contained, we provide many details in our proofs, recall all the prerequisitesfrom the theory of quasicategories, and prove some of those aswell. For instance, we develop the expected facts about (weak) adjunctionsbetween quasicategories and (weak) adjunctions between simplicialcategories.• “Equivariant principal bundles and their classifying spaces”In the paper [76] Uribe and I consider Γ-equivariant principal G-bundlesover proper Γ-CW -complexes with prescribed family of local representations.We construct and analyze their classifying spaces for locally compact,second countable topological groups with finite covering dimensionΓ and G such that G almost connected.14 Survey articlesWe mention the following survey articles: [15, 42, 43, 45, 46, 51, 53, 54,56, 57, 67].8

[13] P. V. M. Blagojević, W. Lück, and G. M. Ziegler. On highly regular embeddings.arXiv:1305.7483v1 [math.AT], 2013.[14] J. F. Davis and W. Lück. The topological K-theory of certain crystallographicgroups. Journal of Non-Commutative Geometry, 7:373–431, 2013.[15] J. Dubois, S. Friedl, and W. Lück. Three flavors of twisted knot invariants.in preparation, 2014.[16] S. Echterhoff, W. Lück, C. Philipps, and S. Walters. The structure ofcrossed products of irrational rotation algebras by finite subgroups ofsl 2 (Ϝ). Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik,Heft 435, Münster, arXiv:math.OA/0609784, to appear in Crelle’sJournal für reine und angewandte Mathematik, 2006.[17] M. Ershov and W. Lück. The first l 2 -Betti number and approximation inarbitrary characteristics. arXiv:1206.0474 [math.GR], to appear in Documenta,2012.[18] F. T. Farrell, W. Lück, and W. Steimle. Obstructions to fibering a manifold.Geom. Dedicata, 148:35–69, 2010.[19] T. Fiore and W. Lück. Waldhausen additivity: Classical and quasicategorical.Preprint, arXiv:1207.6613v1 [math.AT], 2012.[20] T. M. Fiore, W. Lück, and R. Sauer. Euler characteristics of categories andhomotopy colimits. Doc. Math., 16:301–354, 2011.[21] T. M. Fiore, W. Lück, and R. Sauer. Finiteness obstructions and Eulercharacteristics of categories. Adv. Math., 226(3):2371–2469, 2011.[22] M. Joachim and W. Lück. Topological K-(co-)homology of classifyingspaces of discrete groups. A&GT, 13:1–34, 2013.[23] H. Kammeyer, W. Lück, and H. Rüping. The Farrell-Jones Conjecturefor arbitrary lattices in virtually connected Lie groups. Preprint,arXiv:1401.0876 [math.KT], 2014.[24] H. Koch and W. Lück. On the spectral density function of the Laplacian of agraph’. Preprint, arXiv:1205.2321v2 [math.CO], to appear in ExpositionesMathematicae, 2012.[25] M. Kreck and W. Lück. The Novikov conjecture: Geometry and algebra,volume 33 of Oberwolfach Seminars. Birkhäuser Verlag, Basel, 2005.[26] M. Kreck and W. Lück. Topological rigidity for non-aspherical manifolds.Pure and Applied Mathematics Quarterly, 5 (3):873–914, 2009. special issuein honor of Friedrich Hirzebruch.[27] M. Kreck, W. Lück, and P. Teichner. Counterexamples to the Kneserconjecture in dimension four. Comment. Math. Helv., 70(3):423–433, 1995.10

[28] M. Kreck, W. Lück, and P. Teichner. Stable prime decompositions of fourmanifolds.In Prospects in topology (Princeton, NJ, 1994), pages 251–269.Princeton Univ. Press, Princeton, NJ, 1995.[29] M. Langer and W. Lück. On the group cohomology of the semi-directproduct Z n ⋊ ρ Z/m and a conjecture of Adem-Ge-Pan-Petrosyan. J. PureAppl. Algebra, 216:1318–1339, 2012.[30] M. Langer and W. Lück. Topological K-theory of the group C ∗ -algebra ofa semi-direct product Z n ⋊ Z/m for a free conjugation action. J. Topol.Anal., 4(2):121–172, 2012.[31] X. Li and W. Lück. K-theory for ring C ∗ -algebras – the case of numberfields with higher roots of unity. Journal of Topology and Analysis 4 (4),pages 449–479, 2012.[32] P. Linnell, W. Lück, and R. Sauer. The limit of F p -Betti numbers of atower of finite covers with amenable fundamental groups. Proc. Amer.Math. Soc., 139(2):421–434, 2011.[33] J. Lott and W. Lück. L 2 -topological invariants of 3-manifolds. Invent.Math., 120(1):15–60, 1995.[34] W. Lück. Transformation groups and algebraic K-theory, volume 1408 ofLecture Notes in Mathematics. Springer-Verlag, Berlin, 1989.[35] W. Lück. Analytic and topological torsion for manifolds with boundaryand symmetry. J. Differential Geom., 37(2):263–322, 1993.[36] W. Lück. Approximating L 2 -invariants by their finite-dimensional analogues.Geom. Funct. Anal., 4(4):455–481, 1994.[37] W. Lück. L 2 -Betti numbers of mapping tori and groups. Topology,33(2):203–214, 1994.[38] W. Lück. L 2 -torsion and 3-manifolds. In Low-dimensional topology(Knoxville, TN, 1992), pages 75–107. Internat. Press, Cambridge, MA,1994.[39] W. Lück. Dimension theory of arbitrary modules over finite von Neumannalgebras and L 2 -Betti numbers. I. Foundations. J. Reine Angew. Math.,495:135–162, 1998.[40] W. Lück. Dimension theory of arbitrary modules over finite von Neumannalgebras and L 2 -Betti numbers. II. Applications to Grothendieck groups,L 2 -Euler characteristics and Burnside groups. J. Reine Angew. Math.,496:213–236, 1998.[41] W. Lück. The type of the classifying space for a family of subgroups. J.Pure Appl. Algebra, 149(2):177–203, 2000.11

[42] W. Lück. L 2 -invariants and their applications to geometry, group theoryand spectral theory. In Mathematics Unlimited—2001 and Beyond, pages859–871. Springer-Verlag, Berlin, 2001.[43] W. Lück. A basic introduction to surgery theory. In F. T. Farrell,L. Göttsche, and W. Lück, editors, High dimensional manifold theory, number9 in ICTP Lecture Notes, pages 1–224. Abdus Salam International Centrefor Theoretical Physics, Trieste, 2002. Proceedings of the summer school“High dimensional manifold theory” in Trieste May/June 2001, Number 1.http://www.ictp.trieste.it/˜pub off/lectures/vol9.html.[44] W. Lück. Chern characters for proper equivariant homology theories andapplications to K- and L-theory. J. Reine Angew. Math., 543:193–234,2002.[45] W. Lück. L 2 -invariants of regular coverings of compact manifolds andCW-complexes. In Handbook of geometric topology, pages 735–817. North-Holland, Amsterdam, 2002.[46] W. Lück. L 2 -Invariants: Theory and Applications to Geometry andK-Theory, volume 44 of Ergebnisse der Mathematik und ihrer Grenzgebiete.3. Folge. A Series of Modern Surveys in Mathematics [Results inMathematics and Related Areas. 3rd Series. A Series of Modern Surveys inMathematics]. Springer-Verlag, Berlin, 2002.[47] W. Lück. The relation between the Baum-Connes conjecture and the traceconjecture. Invent. Math., 149(1):123–152, 2002.[48] W. Lück. Algebraic topology. Homology and manifolds. Vieweg Studium:Aufbaukurs Mathematik. Vieweg, 2005.[49] W. Lück. The Burnside ring and equivariant stable cohomotopy for infinitegroups. Pure Appl. Math. Q., 1(3):479–541, 2005.[50] W. Lück. K- and L-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by Z/4. Geom. Topol., 9:1639–1676 (electronic),2005.[51] W. Lück. Survey on classifying spaces for families of subgroups. In Infinitegroups: geometric, combinatorial and dynamical aspects, volume 248 ofProgr. Math., pages 269–322. Birkhäuser, Basel, 2005.[52] W. Lück. Rational computations of the topological K-theory of classifyingspaces of discrete groups. J. Reine Angew. Math., 611:163–187, 2007.[53] W. Lück. Survey on geometric group theory. Münster J. of Mathematics,1:73–108, 2008.[54] W. Lück. L 2 -invariants from the algebraic point of view. In Geometric andcohomological methods in group theory, volume 358 of London Math. Soc.Lecture Note Ser., pages 63–161. Cambridge Univ. Press, Cambridge, 2009.12

[55] W. Lück. On the classifying space of the family of virtually cyclic subgroupsfor CAT(0)-groups. Münster J. of Mathematics, 2:201–214, 2009.[56] W. Lück. Survey on aspherical manifolds. In A. Ran, H. te Riele, andJ. Wiegerinck, editors, Proceedings of the 5-th European Congress of MathematicsAmsterdam 14 -18 July 2008, pages 53–82. EMS, 2010.[57] W. Lück. K- and L-theory of group rings. In R. Bhatia, editor, Proceedingsof the 26-th ICM in Hyderabad, August 19-27, 2010, volume II, pages 1071–1098. World Scientific, 2011.[58] W. Lück. Approximating L 2 -invariants and homology growth. Geom.Funct. Anal., 23(2):622–663, 2013.[59] W. Lück and I. Madsen. Equivariant L-theory. I. Math. Z., 203(3):503–526,1990.[60] W. Lück and I. Madsen. Equivariant L-theory. II. Math. Z., 204(2):253–268, 1990.[61] W. Lück and D. Meintrup. On the universal space for group actions withcompact isotropy. In Geometry and topology: Aarhus (1998), pages 293–305. Amer. Math. Soc., Providence, RI, 2000.[62] W. Lück and B. Oliver. Chern characters for the equivariant K-theory ofproper G-CW-complexes. In Cohomological methods in homotopy theory(Bellaterra, 1998), pages 217–247. Birkhäuser, Basel, 2001.[63] W. Lück and B. Oliver. The completion theorem in K-theory for properactions of a discrete group. Topology, 40(3):585–616, 2001.[64] W. Lück and D. Osin. Approximating the first L 2 -Betti number of residuallyfinite groups. J. Topol. Anal., 3(2):153–160, 2011.[65] W. Lück and A. A. Ranicki. Surgery transfer. In Algebraic topology andtransformation groups (Göttingen, 1987), pages 167–246. Springer-Verlag,Berlin, 1988.[66] W. Lück and A. A. Ranicki. Surgery obstructions of fibre bundles. J. PureAppl. Algebra, 81(2):139–189, 1992.[67] W. Lück and H. Reich. The Baum-Connes and the Farrell-Jones conjecturesin K- and L-theory. In Handbook of K-theory. Vol. 1, 2, pages 703–842.Springer, Berlin, 2005.[68] W. Lück, H. Reich, and M. Varisco. Commuting homotopy limits andsmash products. K-Theory, 30(2):137–165, 2003. Special issue in honor ofHyman Bass on his seventieth birthday. Part II.[69] W. Lück and M. Rørdam. Algebraic K-theory of von Neumann algebras.K-Theory, 7(6):517–536, 1993.13

[70] W. Lück and D. Rosenthal. On the K- and L-theory of hyperbolic andvirtually finitely generated abelian groups. arXiv:1007.0845 [math.KT], toappear in Forum Math., 2010.[71] W. Lück and M. Rothenberg. Reidemeister torsion and the K-theory ofvon Neumann algebras. K-Theory, 5(3):213–264, 1991.[72] W. Lück, R. Sauer, and C. Wegner. L 2 -torsion, the measure-theoretic determinantconjecture, and uniform measure equivalence. Journal of Topologyand Analysis, 2 (2):145–171, 2010.[73] W. Lück and T. Schick. L 2 -torsion of hyperbolic manifolds of finite volume.GAFA, 9:518–567, 1999.[74] W. Lück and W. Steimle. Delooping K-theory for additive categories.preprint, arXiv:1303.1272 [math.KT], to appear in the Proceedings ofthe Fourth Arolla Conference on Algebraic Topology 2012, ContemporaryMath., 2013.[75] W. Lück and W. Steimle. A twisted Bass-Heller-Swan decomposition for thenon-connective K-theory of additive categories. preprint, arXiv:1309.1353[math.KT], 2013.[76] W. Lück and B. Uribe. Equivariant principal bundles and their classifyingspaces. preprint, arXiv:1304.4862 [math.AT], to appear in AGT, 2013.[77] W. Lück and M. Weiermann. On the classifying space of the family ofvirtually cyclic subgroups. PAMQ, 8(2):497–555, 2012.AddressWolfgang LückMathematisches InstitutRheinische Friedrich-Wilhelms-Universität BonnEndenicher Allee 60, 53115 Bonn, GermanyHIM (Hausdorff Institute for Mathematics)Poppelsdorfer Allee 45, 53115 Bonn, Germanyemail: wolfgang.lueck@him.uni-bonn.deinternet: http://www.him.uni-bonn.de/lueck14