Abstract
Historically, one of the earliest motivations for the development of K-theory was the need to put on a firm algebraic foundation a number of invariants or obstructions that appear in topology. The primary purpose of this chapter is to examine many of these K-theoretic invariants, not from a historical point of view, but rather a posteriori, now that K-theory is a mature subject.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Douglas R. Anderson. Torsion invariants and actions of finite groups. Michigan Math. J., 29(1):27–42, 1982.
Douglas R. Anderson and W. C. Hsiang. The functors K−1 and pseudoisotopies of polyhedra. Ann. of Math. (2), 105(2):201–223, 1977.
Paweł Andrzejewski. Equivariant finiteness obstruction and its geometric applications – a survey. In Algebraic topology, Pozna´n 1989, volume 1474 of Lecture Notes in Math., pages 20–37. Springer-Verlag, Berlin, 1991.
Shôrô Araki and Katsuo Kawakubo. Equivariant s-cobordism theorems. J. Math. Soc. Japan, 40(2):349–367, 1988.
Amir Assadi and William Browder. On the existence and classification of extensions of actions on submanifolds of disks and spheres. Trans. Amer. Math. Soc., 291(2):487–502, 1985.
Jenny A. Baglivo. An equivariant Wall obstruction theory. Trans. Amer. Math. Soc., 256:305–324, 1979.
M. Bökstedt, G. Carlsson, R. Cohen, T. Goodwillie, W.C. Hsiang, and I. Madsen. On the algebraic K-theory of simply connected spaces. Duke Math. J., 84(3):541–563, 1996.
M. Bökstedt, W.C. Hsiang, and I. Madsen. The cyclotomic trace and algebraic K-theory of spaces. Invent. Math., 111(3):465–539, 1993.
Karol Borsuk. On the topology of retracts. Ann. of Math. (2), 48:1082–1094, 1947.
Rufus Bowen. Markov partitions for Axiom A diffeomorphisms. Amer. J. Math., 92:725–747, 1970.
William Browder and Frank Quinn. A surgery theory for G-manifolds and stratified sets. In Manifolds – Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973), pages 27–36. Univ. Tokyo Press, Tokyo, 1975.
D. Burghelea, L. Friedlander, T. Kappeler, and P. McDonald. Analytic and Reidemeister torsion for representations in finite type Hilbert modules. Geom. Funct. Anal., 6(5):751–859, 1996.
D. Burghelea and R. Lashof. Stability of concordances and the suspension homomorphism. Ann. of Math. (2), 105(3):449–472, 1977.
Sylvain Cappell and Shmuel Weinberger. Homology propagation of group actions. Comm. Pure Appl. Math., 40(6):723–744, 1987.
Sylvain Cappell and Shmuel Weinberger. A simple construction of Atiyah-Singer classes and piecewise linear transformation groups. J. Differential Geom., 33(3):731–742, 1991.
Gunnar Carlsson. Bounded K-theory and the assembly map in algebraic K-theory. In Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), volume 227 of London Math. Soc. Lecture Note Ser., pages 5–127. Cambridge Univ. Press, Cambridge, 1995.
Henri Cartan and Samuel Eilenberg. Homological algebra. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1999. With an appendix by David A. Buchsbaum, Reprint of the 1956 original.
Jean Cerf. La stratification naturelle des espaces de fonctions différentiables réelles et le théorème de la pseudo-isotopie. Inst. Hautes Études Sci. Publ. Math., (39):5–173, 1970.
T.A. Chapman. Topological classification of simple homotopy equivalences. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, pages 509–513. Canad. Math. Congress, Montreal, Que., 1975.
T.A. Chapman. Controlled simple homotopy theory and applications, volume 1009 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1983.
T.A. Chapman and Steve Ferry. Approximating homotopy equivalences by homeomorphisms. Amer. J. Math., 101(3):583–607, 1979.
Jeff Cheeger. Analytic torsion and the heat equation. Ann. of Math. (2), 109(2):259–322, 1979.
Shiing Shen Chern and James Simons. Some cohomology classes in principal fiber bundles and their application to riemannian geometry. Proc. Nat. Acad. Sci. U.S.A., 68:791–794, 1971.
Shiing Shen Chern and James Simons. Characteristic forms and geometric invariants. Ann. of Math. (2), 99:48–69, 1974.
Marshall M. Cohen. A course in simple-homotopy theory. Springer-Verlag, New York, 1973. Graduate Texts in Mathematics, Vol. 10.
J.F. Davis and R.J. Milgram. A survey of the spherical space form problem, volume 2 of Mathematical Reports. Harwood Academic Publishers, Chur, 1985.
Pierre Deligne and Dennis Sullivan. Fibrés vectoriels complexes à groupe structural discret. C. R. Acad. Sci. Paris Sér. A-B, 281(24):Ai, A1081–A1083, 1975.
Bjørn Ian Dundas. Relative K-theory and topological cyclic homology. Acta Math., 179(2):223–242, 1997.
Johan L. Dupont. Curvature and characteristic classes. Springer-Verlag, Berlin, 1978. Lecture Notes in Mathematics, Vol. 640.
F.T. Farrell and L.E. Jones. A topological analogue of Mostow’s rigidity theorem. J. Amer. Math. Soc., 2(2):257–370, 1989.
F.T. Farrell and J.B. Wagoner. Algebraic torsion for infinite simple homotopy types. Comment. Math. Helv., 47:502–513, 1972.
Steve Ferry. Homotoping ε-maps to homeomorphisms. Amer. J. Math., 101(3):567–582, 1979.
Steve Ferry and Andrew Ranicki. A survey of Wall’s finiteness obstruction. In Surveys on surgery theory, Vol. 2, pages 63–79. Princeton Univ. Press, Princeton, NJ, 2001.
Steven C. Ferry and Erik K. Pedersen. Epsilon surgery theory. In Novikov conjectures, index theorems and rigidity, Vol. 2 (Oberwolfach, 1993), volume 227 of London Math. Soc. Lecture Note Ser., pages 167–226. Cambridge Univ. Press, Cambridge, 1995.
A.E. Hatcher. Higher simple homotopy theory. Ann. of Math. (2), 102(1):101–137, 1975.
A.E. Hatcher. Concordance spaces, higher simple-homotopy theory, and applications. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, pages 3–21. Amer. Math. Soc., Providence, R.I., 1978.
Allen Hatcher and John Wagoner. Pseudo-isotopies of compact manifolds. Société Mathématique de France, Paris, 1973. With English and French prefaces, Astérisque, No. 6.
Henning Hauschild. Äquivariante Whiteheadtorsion. Manuscripta Math., 26(1-2):63–82, 1978/79.
Jean-Claude Hausmann. Algebraic K-theory and flat manifolds. In Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), volume 763 of Lecture Notes in Math., pages 212–234. Springer-Verlag, Berlin, 1979.
Jean-Claude Hausmann and Pierre Vogel. The plus construction and lifting maps from manifolds. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, Proc. Sympos. Pure Math., XXXII, pages 67–76. Amer. Math. Soc., Providence, R.I., 1978.
Bruce Hughes and Stratos Prassidis. Control and relaxation over the circle. Mem. Amer. Math. Soc., 145(691):x+96, 2000.
Bruce Hughes and Shmuel Weinberger. Surgery and stratified spaces. In Surveys on surgery theory, Vol. 2, volume 149 of Ann. of Math. Stud., pages 319–352. Princeton Univ. Press, Princeton, NJ, 2001.
C. Bruce Hughes. Bounded homotopy equivalences of Hilbert cube manifolds. Trans. Amer. Math. Soc., 287(2):621–643, 1985.
Thomas Hüttemann, John R. Klein, Wolrad Vogell, Friedhelm Waldhausen, and Bruce Williams. The “fundamental theorem” for the algebraic K-theory of spaces. I. J. Pure Appl. Algebra, 160(1):21–52, 2001.
Thomas Hüttemann, John R. Klein, Wolrad Vogell, Friedhelm Waldhausen, and Bruce Williams. The “fundamental theorem” for the algebraic K-theory of spaces. II. The canonical involution. J. Pure Appl. Algebra, 167(1):53–82, 2002.
Kiyoshi Igusa. On the algebraic K-theory of A ∞-ring spaces. In Algebraic K -theory, Part II lpOberwolfach, 1980), volume 967 of Lecture Notes in Math., pages 146–194. Springer-Verlag, Berlin, 1982.
KiyoshiIgusa.WhathappenstoHatcherandWagoner’sformulasforπ0C(M) when the first Postnikov invariant of M is nontrivial? In Algebraic K -theory, number theory, geometry and analysis (Bielefeld, 1982), volume 1046 of Lecture Notes in Math., pages 104–172. Springer-Verlag, Berlin, 1984.
Kiyoshi Igusa. The stability theorem for smooth pseudoisotopies. K -Theory, 2(1-2):vi+355, 1988.
Kiyoshi Igusa. Higher Franz-Reidemeister torsion, volume 31 of AMS/IPStudies in Advanced Mathematics. American Mathematical Society, Providence, RI, 2002.
Sören Illman. Whitehead torsion and group actions. Ann. Acad. Sci. Fenn. Ser. A I Math., (588):45, 1974.
Sören Illman. Actions of compact Lie groups and the equivariant Whitehead group. Osaka J. Math., 23(4):881–927, 1986.
Michel A. Kervaire. Le théorème de Barden-Mazur-Stallings. Comment. Math. Helv., 40:31–42, 1965.
K.H. Kim and F.W. Roush. The Williams conjecture is false for irreducible subshifts. Ann. of Math. (2), 149(2):545–558, 1999.
Robion C. Kirby and Laurence C. Siebenmann. Foundational essays on topological manifolds, smoothings, and triangulations. Princeton University Press, Princeton, N.J., 1977. With notes by John Milnor and Michael Atiyah, Annals of Mathematics Studies, No. 88.
John R. Klein and John Rognes. The fiber of the linearization map A(∗) → K(Z). Topology, 36(4):829–848, 1997.
Wolfgang Lück. The geometric finiteness obstruction. Proc. London Math. Soc. (3), 54(2):367–384, 1987.
Wolfgang Lück and Mikael Rørdam. Algebraic K-theory of von Neumann algebras. K -Theory, 7(6):517–536, 1993.
I. Madsen, C.B. Thomas, and C.T.C. Wall. The topological spherical space form problem. II. Existence of free actions. Topology, 15(4):375–382, 1976.
J.P. May. A concise course in algebraic topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999.
John Milnor. Groups which act on Sn without fixed points. Amer. J. Math., 79:623–630, 1957.
John Milnor. On theexistence of a connection with curvature zero. Comment. Math. Helv., 32:215–223, 1958.
John Milnor. Lectures on the h -cobordism theorem. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton, N.J., 1965.
John Milnor. Whitehead torsion. Bull. Amer. Math. Soc., 72:358–426, 1966.
John Milnor and Oscar Burlet. Torsion et type simple d’homotopie. In Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), pages 12–17. Springer-Verlag, New York, 1970.
Werner Müller. Analytic torsion and R-torsion of Riemannian manifolds. Adv. in Math., 28(3):233–305, 1978.
Robert Oliver. Whitehead groups of finite groups, volume 132 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1988.
Erik K. Pedersen and Charles A. Weibel. A nonconnective delooping of algebraic K-theory. In Algebraic and geometric topology (New Brunswick, N.J., 1983), volume 1126 of Lecture Notes in Math., pages 166–181. Springer-Verlag, Berlin, 1985.
Erik K. Pedersen and Charles A. Weibel. K-theory homology of spaces. In Algebraic topology (Arcata, CA, 1986), volume 1370 of Lecture Notes in Math., pages 346–361. Springer-Verlag, Berlin, 1989.
Erik Kjaer Pedersen. On the K−i-functors. J. Algebra, 90(2):461–475, 1984.
Frank Quinn. Ends of maps. I. Ann. of Math. (2), 110(2):275–331, 1979.
Frank Quinn. Ends of maps. III. Dimensions 4 and 5. J. Differential Geom., 17(3):503–521, 1982.
Frank Quinn. Homotopically stratified sets. J. Amer. Math. Soc., 1(2):441–499, 1988.
D.B. Ray and I. M. Singer. R-torsion and the Laplacian on Riemannian manifolds. Advances in Math., 7:145–210, 1971.
Jonathan Rosenberg. Algebraic K -theory and its applications, volume 147 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994.
C.P.Rourke.Embeddedhandletheory,concordanceandisotopy.In Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969), pages 431–438. Markham, Chicago, Ill., 1970.
Colin Patrick Rourke and Brian Joseph Sanderson. Introduction to piecewiselinear topology. Springer Study Edition. Springer-Verlag, Berlin, 1982. Reprint.
Roland Schwänzl and Ross E. Staffeldt. The approximation theorem and the K-theory of generalized free products. Trans. Amer. Math. Soc., 347(9):3319–3345, 1995.
L.C. Siebenmann. Finding a boundary for an open manifold of dimension ≥ 5. Ph.D. Dissertation. Princeton Univ., Princeton, 1965.
L.C. Siebenmann. Infinite simple homotopy types. Nederl. Akad. Wetensch. Proc. Ser. A 73 = Indag. Math., 32:479–495, 1970.
S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc., 73:747–817, 1967.
John Smillie. Flat manifolds with non-zero Euler characteristics. Comment. Math. Helv., 52(3):453–455, 1977.
P.A. Smith. Stationary points of transformation groups. Proc. Nat. Acad. Sci. U. S. A., 28:293–297, 1942.
Mark Steinberger. The equivariant topological s-cobordism theorem. Invent. Math., 91(1):61–104, 1988.
Mark Steinberger and James West. Equivariant handles in finite group actions. In Geometry and topology (Athens, Ga., 1985), volume 105 of Lecture Notes in Pure and Appl. Math., pages 277–295. Dekker, New York, 1987.
Andrei A. Suslin. Homology of GLn, characteristic classes and Milnor K-theory. In Algebraic K -theory, number theory, geometry and analysis (Bielefeld, 1982), volume 1046 of Lecture Notes in Math., pages 357–375. Springer-Verlag, Berlin, 1984.
Andrei A. Suslin. On the K-theory of local fields. J. Pure Appl. Algebra, 34(2-3):301–318, 1984.
Richard G. Swan. Periodic resolutions for finite groups. Ann. of Math. (2), 72:267–291, 1960.
C.B. Thomas and C.T.C. Wall. The topological spherical space form problem. I. Compositio Math., 23:101–114, 1971.
J.B. Wagoner. Markov partitions and K2. Inst. Hautes Études Sci. Publ. Math., (65):91–129, 1987.
J.B. Wagoner. Higher-dimensional shift equivalence and strong shift equivalence are the same over the integers. Proc. Amer. Math. Soc., 109(2):527–536, 1990.
J.B. Wagoner. Strong shift equivalence theory and the shift equivalence problem. Bull. Amer. Math. Soc. (N.S.), 36(3):271–296, 1999.
J.B. Wagoner. Strong shift equivalence and K2 of the dual numbers. J. Reine Angew. Math., 521:119–160, 2000. With an appendix by K.H. Kim and F. W. Roush.
Friedhelm Waldhausen. Algebraic K-theory of topological spaces. I. In Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pages 35–60. Amer. Math. Soc., Providence, R.I., 1978.
Friedhelm Waldhausen. Algebraic K-theory of topological spaces. II. In Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ. Aarhus, Aarhus, 1978), pages 356–394. Springer-Verlag, Berlin, 1979.
Friedhelm Waldhausen. Algebraic K-theory of spaces, a manifold approach. In Current trends in algebraic topology, Part 1 (London, Ont., 1981), pages 141–184. Amer. Math. Soc., Providence, R.I., 1982.
Friedhelm Waldhausen. Operations in the algebraic K-theory of spaces. In Algebraic K -theory, Part II (Oberwolfach, 1980), volume 967 of Lecture Notes in Math., pages 390–409. Springer-Verlag, Berlin, 1982.
Friedhelm Waldhausen. Algebraic K-theory of spaces, localization, and the chromatic filtration of stable homotopy. In Algebraic topology, Aarhus 1982 (Aarhus, 1982), pages 173–195. Springer-Verlag, Berlin, 1984.
Friedhelm Waldhausen. Algebraic K-theory of spaces. In Algebraic and geometric topology(New Brunswick, N.J., 1983), pages 318–419. Springer-Verlag, Berlin, 1985.
Friedhelm Waldhausen. Algebraic K-theory of spaces, concordance, and stable homotopy theory. In Algebraic topology and algebraic K -theory (Princeton, N.J., 1983), pages 392–417. Princeton Univ. Press, Princeton, NJ, 1987.
C.T.C. Wall. Finiteness conditions for CW-complexes. Ann. of Math. (2), 81:56–69, 1965.
C.T.C. Wall. Finiteness conditions for CW complexes. II. Proc. Roy. Soc. Ser. A, 295:129–139, 1966.
Shmuel Weinberger. Constructions of group actions: a survey of some recent developments. In Group actions on manifolds (Boulder, Colo., 1983), volume 36 of Contemp. Math., pages 269–298. Amer. Math. Soc., Providence, RI, 1985.
Shmuel Weinberger. The topological classification of stratified spaces. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1994.
R.F. Williams. Classification of subshifts of finite type. Ann. of Math. (2), 98:120–153; errata, ibid. (2) 99 (1974), 380–381, 1973.
Joseph A. Wolf. Spaces of constant curvature. Publish or Perish Inc., Houston, TX, fifth edition, 1984.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this entry
Cite this entry
Rosenberg, J. (2005). K-Theory and Geometric Topology. In: Friedlander, E., Grayson, D. (eds) Handbook of K-Theory. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27855-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-27855-9_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-23019-9
Online ISBN: 978-3-540-27855-9
eBook Packages: Mathematics and StatisticsReference Module Computer Science and Engineering