Geodesic
On the Farrell–Jones conjecture for Waldhausen’s A–theory
Enkelmann, Nils-Edvin1 ; Lück, Wolfgang1 ; Pieper, Malte1 ; Ullmann, Mark2 ; Winges, Christoph3
1 Mathematisches Institut, Rheinische Wilhelms-Universität Bonn, Bonn, Germany
2 Institut für Mathematik, FU Berlin, Berlin, Germany
3 Max-Planck-Institut für Mathematik, Bonn, Germany
Geometry & topology, Tome 22 (2018) no. 6, pp. 3321-3394.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We prove the Farrell–Jones conjecture for (nonconnective) A–theory with coefficients and finite wreath products for hyperbolic groups, CAT(0)–groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudoisotopies in the topological, piecewise linear and smooth categories.

DOI : 10.2140/gt.2018.22.3321
Classification : 19D10, 57Q10, 57Q60
Keywords: Farrell–Jones conjecture, aspherical closed manifolds, $A$–theory, Whitehead spaces, spaces of stable pseudoisotopies, spaces of stable $h$–cobordisms

Enkelmann, Nils-Edvin 1 ; Lück, Wolfgang 1 ; Pieper, Malte 1 ; Ullmann, Mark 2 ; Winges, Christoph 3

1 Mathematisches Institut, Rheinische Wilhelms-Universität Bonn, Bonn, Germany
2 Institut für Mathematik, FU Berlin, Berlin, Germany
3 Max-Planck-Institut für Mathematik, Bonn, Germany 
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Enkelmann, Nils-Edvin; Lück, Wolfgang; Pieper, Malte; Ullmann, Mark; Winges, Christoph. On the Farrell–Jones conjecture for Waldhausen’s A–theory. Geometry & topology, Tome 22 (2018) no. 6, pp. 3321-3394. doi : 10.2140/gt.2018.22.3321. http://geodesic.mathdoc.fr/articles/10.2140/gt.2018.22.3321/

[1] A Bartels, On proofs of the Farrell–Jones conjecture, from: "Topology and geometric group theory" (editors M W Davis, J Fowler, J F Lafont, I J Leary), Springer Proc. Math. Stat. 184, Springer (2016) 1 | DOI

[2] A Bartels, S Echterhoff, W Lück, Inheritance of isomorphism conjectures under colimits, from: "–theory and noncommutative geometry" (editors G Cortiñas, J Cuntz, M Karoubi, R Nest, C A Weibel), Eur. Math. Soc. (2008) 41 | DOI

[3] A Bartels, F T Farrell, W Lück, The Farrell–Jones conjecture for cocompact lattices in virtually connected Lie groups, J. Amer. Math. Soc. 27 (2014) 339 | DOI

[4] A Bartels, T Farrell, L Jones, H Reich, On the isomorphism conjecture in algebraic K–theory, Topology 43 (2004) 157 | DOI

[5] A Bartels, W Lück, The Borel conjecture for hyperbolic and CAT(0)–groups, Ann. of Math. 175 (2012) 631 | DOI

[6] A Bartels, W Lück, H Reich, The K–theoretic Farrell–Jones conjecture for hyperbolic groups, Invent. Math. 172 (2008) 29 | DOI

[7] A Bartels, W Lück, H Reich, H Rüping, K- and L–theory of group rings over GLn(Z), Publ. Math. Inst. Hautes Études Sci. 119 (2014) 97 | DOI

[8] A Berglund, I Madsen, Homological stability of diffeomorphism groups, Pure Appl. Math. Q. 9 (2013) 1 | DOI

[9] A Berglund, I Madsen, Rational homotopy theory of automorphisms of manifolds, preprint (2014)

[10] A Borel, Cohomologie réelle stable de groupes S–arithmétiques classiques, C. R. Acad. Sci. Paris Sér. A-B 274 (1972)

[11] D Burghelea, R Lashof, The homotopy type of the space of diffeomorphisms, I, Trans. Amer. Math. Soc. 196 (1974) 1 | DOI

[12] D Burghelea, R Lashof, The homotopy type of the space of diffeomorphisms, II, Trans. Amer. Math. Soc. 196 (1974) 37 | DOI

[13] J F Davis, W Lück, Spaces over a category and assembly maps in isomorphism conjectures in K– and L–theory, –Theory 15 (1998) 201 | DOI

[14] N E Enkelmann, Pseudoisotopy and automorphism groups of manifolds, PhD Thesis, in preparation

[15] F T Farrell, The Borel conjecture, from: "Topology of high-dimensional manifolds, I" (editors F T Farrell, L Göttsche, W Lück), ICTP Lect. Notes 9, Abdus Salam Int. Cent. Theoret. Phys. (2002) 225

[16] F T Farrell, W C Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, from: "Algebraic and geometric topology, I" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 325 | DOI

[17] F T Farrell, L E Jones, Classical aspherical manifolds, 75, Amer. Math. Soc. (1990) | DOI

[18] F T Farrell, L E Jones, Computations of stable pseudoisotopy spaces for aspherical manifolds, from: "Algebraic topology" (editors S Jackowski, B Oliver, K Pawałowski), Lecture Notes in Math. 1474, Springer (1991) 59 | DOI

[19] F T Farrell, L E Jones, Rigidity in geometry and topology, from: "Proceedings of the International Congress of Mathematicians" (editor I Satake), Math. Soc. Japan (1991) 653

[20] F T Farrell, L E Jones, Stable pseudoisotopy spaces of compact non-positively curved manifolds, J. Differential Geom. 34 (1991) 769 | DOI

[21] F T Farrell, L E Jones, Isomorphism conjectures in algebraic K–theory, J. Amer. Math. Soc. 6 (1993) 249 | DOI

[22] R Fritsch, R A Piccinini, Cellular structures in topology, 19, Cambridge Univ. Press (1990) | DOI

[23] S Galatius, O Randal-Williams, Detecting and realising characteristic classes of manifold bundles, from: "Algebraic topology: applications and new directions" (editors U Tillmann, S Galatius, D Sinha), Contemp. Math. 620, Amer. Math. Soc. (2014) 99 | DOI

[24] S Galatius, O Randal-Williams, Stable moduli spaces of high-dimensional manifolds, Acta Math. 212 (2014) 257 | DOI

[25] S Galatius, O Randal-Williams, Abelian quotients of mapping class groups of highly connected manifolds, Math. Ann. 365 (2016) 857 | DOI

[26] S Galatius, O Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds, II, Ann. of Math. 186 (2017) 127 | DOI

[27] S Galatius, O Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds, I, J. Amer. Math. Soc. 31 (2018) 215 | DOI

[28] J Grunewald, J R Klein, T Macko, Operations on the A-theoretic nil-terms, J. Topol. 1 (2008) 317 | DOI

[29] L Hesselholt, On the Whitehead spectrum of the circle, from: "Algebraic topology" (editors N A Baas, E M Friedlander, B Jahren, P A Østvær), Abel Symp. 4, Springer (2009) 131 | DOI

[30] T Hüttemann, J R Klein, W Vogell, F Waldhausen, B Williams, The “fundamental theorem” for the algebraic K–theory of spaces, I, J. Pure Appl. Algebra 160 (2001) 21 | DOI

[31] T Hüttemann, J R Klein, W Vogell, F Waldhausen, B Williams, The “fundamental theorem” for the algebraic K–theory of spaces, II : The canonical involution, J. Pure Appl. Algebra 167 (2002) 53 | DOI

[32] H Kammeyer, W Lück, H Rüping, The Farrell–Jones conjecture for arbitrary lattices in virtually connected Lie groups, Geom. Topol. 20 (2016) 1275 | DOI

[33] D Kasprowski, M Ullmann, C Wegner, C Winges, The A–theoretic Farrell–Jones conjecture for virtually solvable groups, Bull. Lond. Math. Soc. 50 (2018) 219 | DOI

[34] P Kühl, Isomorphismusvermutungen und 3–Mannigfaltigkeiten, preprint (2009)

[35] W Lück, The transfer maps induced in the algebraic K0– and K1–groups by a fibration, I, Math. Scand. 59 (1986) 93 | DOI

[36] W Lück, Chern characters for proper equivariant homology theories and applications to K- and L–theory, J. Reine Angew. Math. 543 (2002) 193 | DOI

[37] W Lück, H Reich, The Baum–Connes and the Farrell–Jones conjectures in K– and L–theory, from: "Handbook of –theory" (editors E M Friedlander, D R Grayson), Springer (2005) 703 | DOI

[38] W Lück, W Steimle, Splitting the relative assembly map, nil-terms and involutions, Ann. K–Theory 1 (2016) 339 | DOI

[39] W Lück, M Weiermann, On the classifying space of the family of virtually cyclic subgroups, Pure Appl. Math. Q. 8 (2012) 497 | DOI

[40] I Madsen, M Weiss, The stable mapping class group and stable homotopy theory, from: "European Congress of Mathematics" (editor A Laptev), Eur. Math. Soc. (2005) 283 | DOI

[41] I Madsen, M Weiss, The stable moduli space of Riemann surfaces : Mumford’s conjecture, Ann. of Math. 165 (2007) 843 | DOI

[42] M Pieper, Assembly maps and pseudoisotopy functors, PhD thesis, in preparation

[43] F Quinn, Ends of maps, I, Ann. of Math. 110 (1979) 275 | DOI

[44] C P Rourke, B J Sanderson, Block bundles, I, Ann. of Math. 87 (1968) 1 | DOI

[45] S K Roushon, The Farrell–Jones isomorphism conjecture for 3–manifold groups, J. K–Theory 1 (2008) 49 | DOI

[46] H Rüping, The Farrell–Jones conjecture for S–arithmetic groups, J. Topol. 9 (2016) 51 | DOI

[47] M Ullmann, C Winges, On the Farrell–Jones conjecture for algebraic K–theory of spaces : the Farrell–Hsiang method, preprint (2015)

[48] W Vogell, The canonical involution on the algebraic K–theory of spaces, from: "Algebraic topology, Aarhus 1982" (editors I Madsen, B Oliver), Lecture Notes in Math. 1051, Springer (1984) 156 | DOI

[49] W Vogell, Algebraic K–theory of spaces, with bounded control, Acta Math. 165 (1990) 161 | DOI

[50] W Vogell, Boundedly controlled algebraic K–theory of spaces and its linear counterparts, J. Pure Appl. Algebra 76 (1991) 193 | DOI

[51] R M Vogt, Homotopy limits and colimits, Math. Z. 134 (1973) 11 | DOI

[52] F Waldhausen, Algebraic K–theory of topological spaces, I, from: "Algebraic and geometric topology, I" (editor R J Milgram), Proc. Sympos. Pure Math. XXXII, Amer. Math. Soc. (1978) 35 | DOI

[53] F Waldhausen, Algebraic K–theory of spaces, from: "Algebraic and geometric topology" (editors A Ranicki, N Levitt, F Quinn), Lecture Notes in Math. 1126, Springer (1985) 318 | DOI

[54] F Waldhausen, B Jahren, J Rognes, Spaces of PL manifolds and categories of simple maps, 186, Princeton Univ. Press (2013) | DOI

[55] C Wegner, The K–theoretic Farrell–Jones conjecture for CAT(0)–groups, Proc. Amer. Math. Soc. 140 (2012) 779 | DOI

[56] C Wegner, The Farrell–Jones conjecture for virtually solvable groups, J. Topol. 8 (2015) 975 | DOI

[57] M Weiss, B Williams, Automorphisms of manifolds and algebraic K–theory, I, –Theory 1 (1988) 575 | DOI

[58] M Weiss, B Williams, Assembly, from: "Novikov conjectures, index theorems and rigidity, II" (editors S C Ferry, A Ranicki, J Rosenberg), London Math. Soc. Lecture Note Ser. 227, Cambridge Univ. Press (1995) 332 | DOI

[59] M Weiss, B Williams, Automorphisms of manifolds, from: "Surveys on surgery theory, II" (editors S Cappell, A Ranicki, J Rosenberg), Ann. of Math. Stud. 149, Princeton Univ. Press (2001) 165

[60] B Williams, Bivariant Riemann Roch theorems, from: "Geometry and topology" (editors K Grove, I H Madsen, E K Pedersen), Contemp. Math. 258, Amer. Math. Soc. (2000) 377 | DOI

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