BIBLIOGRAFI FOR MA422, VÅREN 1998

ALGEBRAISK K-TEORI OG MANGFOLDIGHETER

John Rognes

17 mars 1998

For homologisk algebra, algebraisk K-teori, og historisk bakgrunn om homologisk algebra, se Weibels bøker [We1] og [We2], og artikkel [We3].

For generell bakgrunn om glatte mangfoldigheter, se Spivak [Sp]. For mer om transversalitet, se Guillemin og Pollack [GuPo]. De tre første kapitlene i Milnor og Stasheff [MiSt] handler også om glatte mangfoldigheter, tangentbunter osv.

For Morse teori, hankelegemer og h-kobordismeteoremet for glatte mangfoldigheter, se Milnors bøker [Mi63] og [Mi65], eller Kosinski [Ko].

For polyhedre og simplisielle komplekser se kapittel 3 i Spanier [Sp]. For mer om PL mangfoldigheter, se Cohen [Co] eller Rourke og Sanderson [RoSa]. Disse dekker også h-kobordismeteoremet for PL mangfoldigheter.

For simplisielle mengder, se Mays bok [Ma], Lamotkes bok [La] eller Goerss og Jardines elektronisk tilgjengelige bok [GoJa].

For sammenhengen mellom algebraisk K-teori og geometrisk topologi, see artiklene til Hatcher, Waldhausen, Brumfield, Madsen, Milgram og Burghelea i boken [AGT].

For kategorier, nerver, klassifiserende rom og spektra, see Segal [Se1], [Se2]. For Q-konstruksjonen av algebraisk K-teori, se Quillen [Q]. For trianguleringer og glattinger se Kirby og Siebenmanns bok [KS].

Referanser

[AGT] R.J. Milgram (ed.), Algebraic and geometric topology, Stanford 1976, Proceedings of Symposia in Pure Mathematics, vol. 32, American Mathematical Society (1980).

[BöWa] M. Bökstedt and F. Waldhausen, The map BSG → A(∗) → QS^{0}, Algebraic topology and algebraic K-theory, Proc. Conf., Princeton, NJ (USA) 1983, Ann. Math. Stud., vol. 113, 1987, pp. 418–431.

[BL1] D. Burghelea and R. Lashof, The homotopy type of the space of diffeomorphisms. I, Trans. Amer. Math. Soc. 196 (1974), 1–36.

[BL2] D. Burghelea and R. Lashof, The homotopy type of the space of diffeomorphisms. II, Trans. Amer. Math. Soc. 196 (1974), 37–50.

[Co] M.M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, vol. 10, Springer-Verlag, 1973.

[GoJa] P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, http://www.math.uwo.ca/ ∼jardine/papers/simp-sets/, 1997.

[GuPo] V. Guillemin and A. Pollack, Differential Topology, Prentice Hall, 1973.

[H1] A. Hatcher, Higher simple homotopy theory, Ann. of Math. 102 (1975), 101–137.

Typeset by AMS-TEX [H2] A. Hatcher, Concordance spaces, higher simple-homotopy theory, and applications, Al-

gebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., vol. 22, Part 1, 1978, pp. 3–21.

[KS] R.C. Kirby and L.C. Siebenmann, Foundational essays on topological manifolds, smoothings and triangulations, Annals of Mathematics Studies, vol. 88, Princeton University Press, 1977.

[Ko] A. Kosinski, Differential Manifolds, Pure and Applied Mathematics, vol. 138, Academic Press, 1993.

[La] K. Lamotke, Semisimpliziale algebraische Topologie (German), Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, vol. 147, Springer-Verlag, 1968.

[Ma] J.P. May, Simplicial Objects in Algebraic Topology, Van Nostrand mathematical studies, vol. 11, Van Nostrand, 1967.

[Mi63] J.W. Milnor, Morse Theory, Annals of Mathematics Studies, vol. 51, Princeton University Press, 1963.

[Mi65] J.W. Milnor, Lectures on the h-cobordism theorem, Princeton Mathematical Notes, Princeton University Press, 1965.

[Mi66] J.W. Milnor, Whitehead torsion, Bull. Am. Math. Soc. 72 (1966), 358–426.

[MiSt] J.W. Milnor and J.D. Stasheff, Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, 1974.

[Q] D. Quillen, Higher algebraic K-theory. I, Algebr. K-Theory I, Proc. Conf. Battelle Inst. 1972, Lecture Notes Math., vol. 341, 1973, pp. 85–147.

[RoSa] C.P. Rourke and B.J. Sanderson, Introduction to piecewise-linear topology, Springer Study Edition, Springer-Verlag, 1982.

[Se1] G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Etud. Sci., Publ. Math. 34 (1968), 105–112.

[Se2] G. Segal, Categories and cohomology theories, Topology 13 (1974), 293–312.

[Sp] M. Spivak, Differential Geometry, vol. I, Publish or Perish, 1979.

[Wa78] F. Waldhausen, Algebraic K-theory of topological spaces. I, Algebr. geom. Topol., Stanford/Calif. 1976, Proc. Symp. Pure Math., vol. 22, Part 1, 1978, pp. 35–60.

[Wa79] F. Waldhausen, Algebraic K-theory of topological spaces. II, Algebraic topology, Proc. Symp., Aarhus 1978, Lect. Notes Math., vol. 763, Springer-Verlag, 1979, pp. 356–394.

[Wa82a] F. Waldhausen Friedhelm, Operations in the algebraic K-theory of spaces, Algebraic K-theory, Proc. Conf., Oberwolfach 1980, Part II, Lect. Notes Math., vol. 967, Springer-Verlag, 1982, pp. 390–409.

[Wa82b] F. Waldhausen, Algebraic K-theory of spaces, a manifold approach, Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, Part 1, 1982, pp. 141– 184.

[Wa84] F. Waldhausen, Algebraic K-theory of spaces, localization, and the chromatic filtration of stable homotopy, Algebraic topology, Proc. Conf., Aarhus 1982, Lect. Notes Math., vol. 1051, Springer-Verlag, 1984, pp. 173–195.

[Wa85] F. Waldhausen, Algebraic K-theory of spaces, Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lect. Notes Math., vol. 1126, Springer-Verlag, 1985, pp. 318–419.

[Wa87a] F. Waldhausen, Algebraic K-theory of spaces, concordance, and stable homotopy theory, Algebraic topology and algebraic K-theory, Proc. Conf., Princeton, NJ (USA) 1983, Ann. Math. Stud., vol. 113, 1987, pp. 392–417.

[Wa87b] F. Waldhausen, An outline of how manifolds relate to algebraic K-theory, Homotopy theory, Proc. Symp., Durham/Engl. 1985, Lond. Math. Soc. Lect. Note Ser., vol. 117, 1987, pp. 239–247.

[Wa96] F. Waldhausen, On the construction of the Kan loop group, http://www.mathematik. uni-bielefeld.de/documenta, Doc. Math. 1 (1996), 121–126.

[We1] C. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge Univ. Press, 1995.

[We2] C. Weibel, An introduction to algebraic K-theory, http://math.rutgers.edu:80/∼weibel/ Kbook.html.

[We3] C. Weibel, A history of homological algebra, http://math.rutgers.edu:80/∼weibel/ history.dvi.

BIBLIOGRAFI FOR MA422 3

[WW1] M. Weiss and B. Williams, Automorphisms of manifolds and algebraic K-theory: I, K-Theory 1 (1988), 575–626.