×

zbMATH — the first resource for mathematics

On the algebraic \(K\)-theory of simply connected spaces. (English) Zbl 0867.19003
This paper gives an interpretation of Waldhausen’s (reduced) functor \(\tilde A(X)\) in terms of more familiar objects of algebraic topology. The main ingredients are the cyclotomic trace invariant of M. Bökstedt, W. C. Hsiang and I. Madsen [Invent. Math. 111, No. 3, 465-539 (1993; Zbl 0804.55004)] and the calculus of functors of T. G. Goodwillie [“The differential calculus of homotopy functors”, in: Proc. Int. Congr. Math., Kyoto 1990, Vol. I, 621-630 (1991; Zbl 0759.55011)].
The main result of the paper is that after \(p\)-completion \(\tilde A(X)\) agrees with \(\widetilde{TC}(X;p)\) for a simply connected space \(X\) of finite type. Moreover, the authors give a description of the \(p\)-completion of \(\widetilde{TC}(X;p)\). This description is then used to describe the \(p\)-completion of \(\widetilde{Wh}(X)\). Applications to the homotopy type of the space of smooth pseudo-isotopies of a manifold and integral versions of the results are discussed.
Reviewer: A.Cap (Wien)

MSC:
19D10 Algebraic \(K\)-theory of spaces
55P42 Stable homotopy theory, spectra
55P65 Homotopy functors in algebraic topology
57R52 Isotopy in differential topology
55N15 Topological \(K\)-theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] J. F. Adams, Prerequisites (on equivariant stable homotopy) for Carlsson’s lecture , Algebraic topology, Aarhus 1982 (Aarhus, 1982), Lecture Notes in Math., vol. 1051, Springer, Berlin, 1984, pp. 483-532. · Zbl 0553.55010 · doi:10.1007/BFb0075584
[2] C. F. Bödigheimer and I. Madsen, Homotopy quotients of mapping spaces and their stable splitting , Quart. J. Math. Oxford Ser. (2) 39 (1988), no. 156, 401-409. · Zbl 0664.55009 · doi:10.1093/qmath/39.4.401
[3] M. Bökstedt, Topological Hochschild homology , to appear in Topology.
[4] M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic \(K\)-theory of spaces , Invent. Math. 111 (1993), no. 3, 465-539. · Zbl 0804.55004 · doi:10.1007/BF01231296 · eudml:144086
[5] A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations , Lecture Notes in Mathematics, vol. 304, Springer-Verlag, Berlin, 1972. · Zbl 0259.55004
[6] D. Burghelea, Cyclic homology and the algebraic \(K\)-theory of spaces. I , Applications of algebraic \(K\)-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983), Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 89-115. · Zbl 0615.55009
[7] G. Carlsson and R. Cohen, The cyclic groups and the free loop space , Comment. Math. Helv. 62 (1987), no. 3, 423-449. · Zbl 0632.57028 · doi:10.1007/BF02564455 · eudml:140092
[8] G. Carlsson, R. Cohen, T. Goodwillie, and W. Hsiang, The free loop space and the algebraic \(K\)-theory of spaces , \(K\)-Theory 1 (1987), no. 1, 53-82. · Zbl 0649.55001 · doi:10.1007/BF00533987
[9] G. Carlsson, R. Cohen, and W. C. Hsiang, The homotopy type of the space of pseudo-isotopies ,
[10] T. Goodwillie, Calculus. I. The first derivative of pseudoisotopy theory , \(K\)-Theory 4 (1990), no. 1, 1-27. · Zbl 0741.57021 · doi:10.1007/BF00534191
[11] T. Goodwillie, Calculus. II. Analytic functors , \(K\)-Theory 5 (1991/92), no. 4, 295-332. · Zbl 0776.55008 · doi:10.1007/BF00535644
[12] T. Goodwillie, The differential calculus of homotopy functors , Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, pp. 621-630. · Zbl 0759.55011
[13] K. Igusa, The stability theorem for smooth pseudoisotopies , \(K\)-Theory 2 (1988), no. 1-2, vi+355. · Zbl 0691.57011 · doi:10.1007/BF00533643
[14] J. Loday and D. Quillen, Cyclic homology and the Lie algebra homology of matrices , Comment. Math. Helv. 59 (1984), no. 4, 569-591. · Zbl 0565.17006 · doi:10.1007/BF02566367 · eudml:139991
[15] I. Madsen, Algebraic \(K\)-theory and traces , Current developments in mathematics, 1995 (Cambridge, MA), Internat. Press, Cambridge, MA, 1994, pp. 191-321.
[16] D. Sullivan, Genetics of homotopy theory and the Adams conjecture , Ann. of Math. (2) 100 (1974), 1-79. · Zbl 0355.57007 · doi:10.2307/1970841
[17] F. Waldhausen, Algebraic \(K\)-theory of spaces, a manifold approach , Current trends in algebraic topology, Part 1 (London, Ont., 1981), CMS Conf. Proc., vol. 2, Amer. Math. Soc., Providence, R.I., 1982, pp. 141-184. · Zbl 0595.57026
[18] F. Waldhausen, Algebraic \(K\)-theory of spaces, concordance, and stable homotopy theory , Algebraic topology and algebraic \(K\)-theory (Princeton, N.J., 1983), Ann. of Math. Stud., vol. 113, Princeton Univ. Press, Princeton, NJ, 1987, pp. 392-417. · Zbl 0708.19001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.