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Automorphisms of manifolds and algebraic $K$–theory: Part III
About this Title
Michael S. Weiss, Mathematisches Institut, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany and Bruce E. Williams, Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556-5683
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 231, Number 1084
ISBNs: 978-1-4704-0981-4 (print); 978-1-4704-1720-8 (online)
DOI: https://doi.org/10.1090/memo/1084
Published electronically: January 14, 2014
MSC: Primary 57R65, 18F25, secondary, 55R10, 57R22
Table of Contents
Chapters
- 1. Introduction
- 2. Outline of proof
- 3. Visible $L$-theory revisited
- 4. The hyperquadratic $L$–theory of a point
- 5. Excision and restriction in controlled $L$–theory
- 6. Control and visible $L$-theory
- 7. Control, stabilization and change of decoration
- 8. Spherical fibrations and twisted duality
- 9. Homotopy invariant characteristics and signatures
- 10. Excisive characteristics and signatures
- 11. Algebraic approximations to structure spaces: Set-up
- 12. Algebraic approximations to structure spaces: Constructions
- 13. Algebraic models for structure spaces: Proofs
- A. Homeomorphism groups of some stratified spaces
- B. Controlled homeomorphism groups
- C. $K$-theory of pairs and diagrams
- D. Corrections and Elaborations
Abstract
The structure space $\mathcal {S}(M)$ of a closed topological $m$-manifold $M$ classifies bundles whose fibers are closed $m$-manifolds equipped with a homotopy equivalence to $M$. We construct a highly connected map from $\mathcal {S}(M)$ to a concoction of algebraic $L$-theory and algebraic $K$-theory spaces associated with $M$. The construction refines the well-known surgery theoretic analysis of the block structure space of $M$ in terms of $L$-theory.- J. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill.-London, 1974. Chicago Lectures in Mathematics. MR 0402720
- J. F. Adams, Prerequisites (on equivariant stable homotopy theory) for Carlsson’s lecture, Proc. of 1982 Aarhus Symposium on algebraic topology, Lecture Notes in Maths., vol. 1051, Springer, 1984, pp. 483–532.
- Douglas R. Anderson, Francis X. Connolly, Steven C. Ferry, and Erik K. Pedersen, Algebraic $K$-theory with continuous control at infinity, J. Pure Appl. Algebra 94 (1994), no. 1, 25–47. MR 1277522, DOI 10.1016/0022-4049(94)90004-3
- 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
- J. Bryant, S. Ferry, W. Mio, and S. Weinberger, Topology of homology manifolds, Ann. of Math. (2) 143 (1996), no. 3, 435–467. MR 1394965, DOI 10.2307/2118532
- D. Burghelea and R. Lashof, Geometric transfer and the homotopy type of the automorphism groups of a manifold, Trans. Amer. Math. Soc. 269 (1982), no. 1, 1–38. MR 637027, DOI 10.1090/S0002-9947-1982-0637027-4
- Gunnar Carlsson, Equivariant stable homotopy and Segal’s Burnside ring conjecture, Ann. of Math. (2) 120 (1984), no. 2, 189–224. MR 763905, DOI 10.2307/2006940
- Gunnar Carlsson, A survey of equivariant stable homotopy theory, Topology 31 (1992), no. 1, 1–27. MR 1153236, DOI 10.1016/0040-9383(92)90061-L
- Gunnar Carlsson, Erik Kjær Pedersen, and Wolrad Vogell, Continuously controlled algebraic $K$-theory of spaces and the Novikov conjecture, Math. Ann. 310 (1998), no. 1, 169–182. MR 1600085, DOI 10.1007/s002080050143
- W. Dwyer, M. Weiss, and B. Williams, A parametrized index theorem for the algebraic $K$-theory Euler class, Acta Math. 190 (2003), no. 1, 1–104. MR 1982793, DOI 10.1007/BF02393236
- Robert D. Edwards and Robion C. Kirby, Deformations of spaces of imbeddings, Ann. of Math. (2) 93 (1971), 63–88. MR 283802, DOI 10.2307/1970753
- Kiyoshi Igusa, The stability theorem for smooth pseudoisotopies, $K$-Theory 2 (1988), no. 1-2, vi+355. MR 972368, DOI 10.1007/BF00533643
- I. M. James, Fibrewise topology, Cambridge Tracts in Mathematics, vol. 91, Cambridge University Press, Cambridge, 1989. MR 1010230
- John N. Mather, The vanishing of the homology of certain groups of homeomorphisms, Topology 10 (1971), 297–298. MR 288777, DOI 10.1016/0040-9383(71)90022-X
- J. P. May, Equivariant homotopy and cohomology theory, CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. MR 1413302
- Dusa McDuff, The homology of some groups of diffeomorphisms, Comment. Math. Helv. 55 (1980), no. 1, 97–129. MR 569248, DOI 10.1007/BF02566677
- A. S. Miščenko, Homotopy invariants of multiply connected manifolds. III. Higher signatures, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1316–1355 (Russian). MR 0293658
- Andrew Ranicki, The total surgery obstruction, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979, pp. 275–316. MR 561227
- Andrew Ranicki, The algebraic theory of surgery. I. Foundations, Proc. London Math. Soc. (3) 40 (1980), no. 1, 87–192. MR 560997, DOI 10.1112/plms/s3-40.1.87
- Andrew Ranicki, The algebraic theory of surgery, II. Applications to topology, Proc. Lond. Math. Soc. 40 (1980), 193–287.
- A. A. Ranicki, Algebraic $L$-theory and topological manifolds, Cambridge Tracts in Mathematics, vol. 102, Cambridge University Press, Cambridge, 1992. MR 1211640
- A. A. Ranicki, Lower $K$– and $L$–theory, Lond. Math. Soc. Lecture Note Series, vol. 178, Camb. Univ. Press, 1992.
- Julius L. Shaneson, Wall’s surgery obstruction groups for $G\times Z$, Ann. of Math. (2) 90 (1969), 296–334. MR 246310, DOI 10.2307/1970726
- L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972), 123–136; ibid. 47 (1972), 137–163. MR 319207, DOI 10.1007/BF02566793
- Pierre Vogel, Une nouvelle famille de groupes en $L$-théorie algébrique, Algebraic $K$-theory, number theory, geometry and analysis (Bielefeld, 1982) Lecture Notes in Math., vol. 1046, Springer, Berlin, 1984, pp. 385–421 (French). MR 750692, DOI 10.1007/BFb0072033
- Wolrad Vogell, The involution in the algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 277–317. MR 802795, DOI 10.1007/BFb0074448
- Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI 10.1007/BFb0074449
- F. Waldhausen, B. Jahren, and J. Rognes, Spaces of pl manifolds and categories of simple maps, Princeton University Press, to appear.
- C. T. C. Wall, Poincaré complexes. I, Ann. of Math. (2) 86 (1967), 213–245. MR 217791, DOI 10.2307/1970688
- C. T. C. Wall, Surgery on compact manifolds, 2nd ed., Mathematical Surveys and Monographs, vol. 69, American Mathematical Society, Providence, RI, 1999. Edited and with a foreword by A. A. Ranicki. MR 1687388
- Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324
- M. Weiss, Surgery and the generalized Kervaire invariant, Proc. Lond. Math. Soc. 51 (1985), I. 146–192, II. 193–230.
- Michael Weiss, Visible $L$-theory, Forum Math. 4 (1992), no. 5, 465–498. MR 1176883, DOI 10.1515/form.1992.4.465
- Michael Weiss, Hammock localization in Waldhausen categories, J. Pure Appl. Algebra 138 (1999), no. 2, 185–195. MR 1689629, DOI 10.1016/S0022-4049(98)00009-7
- Michael Weiss, Excision and restriction in controlled $K$-theory, Forum Math. 14 (2002), no. 1, 85–119. MR 1880196, DOI 10.1515/form.2002.007
- Michael Weiss and Bruce Williams, Automorphisms of manifolds and algebraic $K$-theory. I, $K$-Theory 1 (1988), no. 6, 575–626. MR 953917, DOI 10.1007/BF00533787
- Michael Weiss and Bruce Williams, Automorphisms of manifolds and algebraic $K$-theory. II, J. Pure Appl. Algebra 62 (1989), no. 1, 47–107. MR 1026874, DOI 10.1016/0022-4049(89)90020-0
- Michael Weiss and Bruce Williams, Assembly, Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993) London Math. Soc. Lecture Note Ser., vol. 227, Cambridge Univ. Press, Cambridge, 1995, pp. 332–352. MR 1388318, DOI 10.1017/CBO9780511629365.014
- Michael Weiss and Bruce Williams, Duality in Waldhausen categories, Forum Math. 10 (1998), no. 5, 533–603. MR 1644309, DOI 10.1515/form.10.5.533
- Michael S. Weiss and Bruce Williams, Products and duality in Waldhausen categories, Trans. Amer. Math. Soc. 352 (2000), no. 2, 689–709. MR 1694381, DOI 10.1090/S0002-9947-99-02552-0
- Michael Weiss and Bruce Williams, Automorphisms of manifolds, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud., vol. 149, Princeton Univ. Press, Princeton, NJ, 2001, pp. 165–220. MR 1818774