A topological analogue of Mostow’s rigidity theorem
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- by F. T. Farrell and L. E. Jones
- J. Amer. Math. Soc. 2 (1989), 257-370
- DOI: https://doi.org/10.1090/S0894-0347-1989-0973309-4
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References
- Douglas R. Anderson and W. C. Hsiang, The functors $K_{-i}$ and pseudo-isotopies of polyhedra, Ann. of Math. (2) 105 (1977), no. 2, 201–223. MR 440573, DOI 10.2307/1970997
- D. V. Anosov, Geodesic flows on closed Riemann manifolds with negative curvature. , Proceedings of the Steklov Institute of Mathematics, No. 90 (1967), American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by S. Feder. MR 0242194
- Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
- R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49. MR 251664, DOI 10.1090/S0002-9947-1969-0251664-4
- Armand Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. (4) 7 (1974), 235–272 (1975). MR 387496, DOI 10.24033/asens.1269
- William Browder and Frank Quinn, A surgery theory for $G$-manifolds and stratified sets, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 27–36. MR 0375348
- Dan Burghelea and Richard Lashof, The homotopy type of the space of diffeomorphisms. I, II, Trans. Amer. Math. Soc. 196 (1974), 1–36; ibid. 196 (1974), 37–50. MR 356103, DOI 10.1090/S0002-9947-1974-0356103-2 —, The homotopy type of the space of diffeomorphisms. II, Trans. Amer. Math. Soc. 196 (1975), 37-50.
- D. Burghelea and R. Lashof, Stability of concordances and the suspension homomorphism, Ann. of Math. (2) 105 (1977), no. 3, 449–472. MR 438365, DOI 10.2307/1970919
- Dan Burghelea, Automorphisms of manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 347–371. MR 520511
- Steve Ferry, Homotoping $\varepsilon$-maps to homeomorphisms, Amer. J. Math. 101 (1979), no. 3, 567–582. MR 533191, DOI 10.2307/2373798
- David Epstein and Michael Shub, Expanding endomorphisms of flat manifolds, Topology 7 (1968), 139–141. MR 227996, DOI 10.1016/0040-9383(68)90022-0
- F. T. Farrell, The obstruction to fibering a manifold over a circle, Indiana Univ. Math. J. 21 (1971/72), 315–346. MR 290397, DOI 10.1512/iumj.1971.21.21024
- F. T. Farrell and W. C. Hsiang, Manifolds with $\pi _{i}=G\times \alpha T$, Amer. J. Math. 95 (1973), 813–848. MR 385867, DOI 10.2307/2373698
- F. T. Farrell and W. C. Hsiang, Rational $L$-groups of Bieberbach groups, Comment. Math. Helv. 52 (1977), no. 1, 89–109. MR 448372, DOI 10.1007/BF02567358
- F. T. Farrell and W. C. Hsiang, The topological-Euclidean space form problem, Invent. Math. 45 (1978), no. 2, 181–192. MR 482771, DOI 10.1007/BF01390272
- F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 325–337. MR 520509 —, On Novikov’s conjecture for non-positively curved manifolds, I, Ann. of Math. (2) 113 (1981), 199-209.
- F. T. Farrell and W. C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups, J. London Math. Soc. (2) 24 (1981), no. 2, 308–324. MR 631942, DOI 10.1112/jlms/s2-24.2.308
- F. T. Farrell and W. C. Hsiang, The stable topological-hyperbolic space form problem for complete manifolds of finite volume, Invent. Math. 69 (1982), no. 1, 155–170. MR 671656, DOI 10.1007/BF01389189
- F. T. Farrell and W. C. Hsiang, Topological characterization of flat and almost flat Riemannian manifolds $M^{n}$ $(n\not =3,\,4)$, Amer. J. Math. 105 (1983), no. 3, 641–672. MR 704219, DOI 10.2307/2374318
- F. T. Farrell and L. E. Jones, Algebraic $K$-theory of spaces stratified fibered over hyperbolic orbifolds, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 15, 5364–5366. MR 851508, DOI 10.1073/pnas.83.15.5364
- F. T. Farrell and L. E. Jones, $K$-theory and dynamics. I, Ann. of Math. (2) 124 (1986), no. 3, 531–569. MR 866708, DOI 10.2307/2007092
- F. T. Farrell and L. E. Jones, Algebraic $K$-theory of discrete subgroups of Lie groups, Proc. Nat. Acad. Sci. U.S.A. 84 (1987), no. 10, 3095–3096. MR 887290, DOI 10.1073/pnas.84.10.3095
- F. T. Farrell and L. E. Jones, $K$-theory and dynamics. II, Ann. of Math. (2) 126 (1987), no. 3, 451–493. MR 916716, DOI 10.2307/1971358
- F. T. Farrell and L. E. Jones, The surgery $L$-groups of poly-(finite or cyclic) groups, Invent. Math. 91 (1988), no. 3, 559–586. MR 928498, DOI 10.1007/BF01388787
- F. T. Farrell and L. E. Jones, Topological rigidity for hyperbolic manifolds, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 1, 277–282. MR 940487, DOI 10.1090/S0273-0979-1988-15640-6
- F. T. Farrell and L. E. Jones, Foliated control theory. I, II, $K$-Theory 2 (1988), no. 3, 357–430. MR 972605, DOI 10.1007/BF00533389 —, Foliated control theory II, $K$-theory 2 (1988), 401-430. —, Splitting over flat and almost flat Riemannian manifolds (in preparation). M. Freedman and F. Quinn, $4$-manifolds, Ann. of Math. Stud. (to appear).
- M. Gromov, Manifolds of negative curvature, J. Differential Geometry 13 (1978), no. 2, 223–230. MR 540941, DOI 10.4310/jdg/1214434487
- A. E. Hatcher, Concordance spaces, higher simple-homotopy theory, and applications, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 3–21. MR 520490
- Allen Hatcher and John Wagoner, Pseudo-isotopies of compact manifolds, Astérisque, No. 6, Société Mathématique de France, Paris, 1973. With English and French prefaces. MR 0353337
- Noel J. Hicks, Notes on differential geometry, Van Nostrand Mathematical Studies, No. 3, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0179691
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
- Kiyoshi Igusa, The stability theorem for smooth pseudoisotopies, $K$-Theory 2 (1988), no. 1-2, vi+355. MR 972368, DOI 10.1007/BF00533643
- Lowell Jones, Patch spaces: a geometric representation for Poincaré spaces, Ann. of Math. (2) 97 (1973), 306–343. MR 315720, DOI 10.2307/1970849 —, The non-simply connected characteristic variety theorem, Proc. Sympos. Pure Math., Vol. 32, Amer. Math. Soc., Providence, RI, 1978, pp. 131-140.
- Lowell Jones, Construction of surgery problems, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 367–391. MR 537742
- G. G. Kasparov, Topological invariants of elliptic operators. I. $K$-homology, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 4, 796–838 (Russian); Russian transl., Math. USSR-Izv. 9 (1975), no. 4, 751–792 (1976). MR 0488027
- G. G. Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR 918241, DOI 10.1007/BF01404917
- Robion C. Kirby and Laurence C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR 0645390, DOI 10.1515/9781400881505 H. Kneser, Geschlossene Flächen in dreidimensionalen Mannigfalligkeiten, Jahresber. Deutsch. Math.-Verein. 38 (1929), 248-260.
- Norman Levitt, Poincaré duality cobordism, Ann. of Math. (2) 96 (1972), 211–244. MR 314059, DOI 10.2307/1970787
- Serge Maumary, Proper surgery groups and Wall-Novikov groups, Algebraic $K$-theory, III: Hermitian $K$-theory and geometric applications (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 343, Springer, Berlin, 1973, pp. 526–539. MR 0377938
- J. Milnor, A unique decomposition theorem for $3$-manifolds, Amer. J. Math. 84 (1962), 1–7. MR 142125, DOI 10.2307/2372800
- John Milnor, On fundamental groups of complete affinely flat manifolds, Advances in Math. 25 (1977), no. 2, 178–187. MR 454886, DOI 10.1016/0001-8708(77)90004-4
- A. S. Miščenko, Infinite-dimensional representations of discrete groups, and higher signatures, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 81–106 (Russian). MR 0362407
- G. D. Mostow, Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 53–104. MR 236383, DOI 10.1007/BF02684590
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
- Andrew J. Nicas, Induction theorems for groups of homotopy manifold structures, Mem. Amer. Math. Soc. 39 (1982), no. 267, vi+108. MR 668807, DOI 10.1090/memo/0267
- Andrew J. Nicas, An infinite family of non-Haken hyperbolic $3$-manifolds with vanishing Whitehead groups, Math. Proc. Cambridge Philos. Soc. 99 (1986), no. 2, 239–246. MR 817665, DOI 10.1017/S030500410006415X
- A. J. Nicas and C. W. Stark, Higher Whitehead groups of certain bundles over Seifert manifolds, Proc. Amer. Math. Soc. 91 (1984), no. 1, 1–5. MR 735551, DOI 10.1090/S0002-9939-1984-0735551-4
- A. J. Nicas and C. W. Stark, Whitehead groups of certain hyperbolic manifolds. II, Combinatorial group theory and topology (Alta, Utah, 1984) Ann. of Math. Stud., vol. 111, Princeton Univ. Press, Princeton, NJ, 1987, pp. 415–431. MR 895625
- Gopal Prasad, Strong rigidity of $\textbf {Q}$-rank $1$ lattices, Invent. Math. 21 (1973), 255–286. MR 385005, DOI 10.1007/BF01418789
- Frank Quinn, A geometric formulation of surgery, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 500–511. MR 0282375
- Frank Quinn, A geometric formulation of surgery, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 500–511. MR 0282375
- Frank Quinn, Ends of maps. I, Ann. of Math. (2) 110 (1979), no. 2, 275–331. MR 549490, DOI 10.2307/1971262
- A. A. Ranicki, Algebraic $L$-theory. I. Foundations, Proc. London Math. Soc. (3) 27 (1973), 101–125. MR 414661, DOI 10.1112/plms/s3-27.1.101 L. C. Siebenmann, The obstruction to finding a boundary for an open manifold of dimension $> 5$, Ph.D. thesis, Princeton Univ., 1965. L. Taylor, Surgery on paracompact manifolds, Ph.D. thesis, Univ. of California at Berkeley, 1972.
- Friedhelm Waldhausen, Algebraic $K$-theory of topological spaces. I, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 35–60. MR 520492
- C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
- Frank Quinn, Ends of maps. II, Invent. Math. 68 (1982), no. 3, 353–424. MR 669423, DOI 10.1007/BF01389410
- F. T. Farrell and L. E. Jones, A topological analogue of Mostow’s rigidity theorem, J. Amer. Math. Soc. 2 (1989), no. 2, 257–370. MR 973309, DOI 10.1090/S0894-0347-1989-0973309-4 —, Negatively curved manifolds with exotic smooth structures (submitted for publication).
Bibliographic Information
- © Copyright 1989 American Mathematical Society
- Journal: J. Amer. Math. Soc. 2 (1989), 257-370
- MSC: Primary 57N15; Secondary 19B28, 22E40, 57R52, 57R67, 57S05, 58F17
- DOI: https://doi.org/10.1090/S0894-0347-1989-0973309-4
- MathSciNet review: 973309