Geometry & Topology

The smooth Whitehead spectrum of a point at odd regular primes

John Rognes

Full-text: Open access

Abstract

Let p be an odd regular prime, and assume that the Lichtenbaum–Quillen conjecture holds for K([1p]) at p. Then the p–primary homotopy type of the smooth Whitehead spectrum Wh() is described. A suspended copy of the cokernel-of-J spectrum splits off, and the torsion homotopy of the remainder equals the torsion homotopy of the fiber of the restricted S1-transfer map t:ΣPS. The homotopy groups of Wh() are determined in a range of degrees, and the cohomology of Wh() is expressed as an A-module in all degrees, up to an extension. These results have geometric topological interpretations, in terms of spaces of concordances or diffeomorphisms of highly connected, high dimensional compact smooth manifolds.

Article information

Source
Geom. Topol., Volume 7, Number 1 (2003), 155-184.

Dates
Received: 30 November 2001
Revised: 7 February 2003
Accepted: 13 March 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883095

Digital Object Identifier
doi:10.2140/gt.2003.7.155

Mathematical Reviews number (MathSciNet)
MR1988283

Zentralblatt MATH identifier
1130.19300

Subjects
Primary: 19D10: Algebraic $K$-theory of spaces
Secondary: 19F27: Étale cohomology, higher regulators, zeta and L-functions [See also 11G40, 11R42, 11S40, 14F20, 14G10] 55P42: Stable homotopy theory, spectra 55Q52: Homotopy groups of special spaces 57R50: Diffeomorphisms 57R80: $h$- and $s$-cobordism

Keywords
algebraic $K$-theory topological cyclic homology Lichtenbaum–Quillen conjecture transfer $h$-cobordism concordance pseudoisotopy

Citation

Rognes, John. The smooth Whitehead spectrum of a point at odd regular primes. Geom. Topol. 7 (2003), no. 1, 155--184. doi:10.2140/gt.2003.7.155. https://projecteuclid.org/euclid.gt/1513883095


Export citation

References

  • J F Adams, Lectures on generalised cohomology , Category Theory, Homology Theory and their Applications, Proc. Conf. Seattle Res. Center Battelle Mem. Inst. 1968, 3 (1969), 1–138
  • J F Adams S B Priddy, Uniqueness of $BSO$ , Math. Proc. Cambridge Philos. Soc., 80 (1976), 475–509
  • S Betley, On the homotopy groups of $A(X)$ , Proc. Amer. Math. Soc., 98 (1986), 495–498
  • S Bloch K Kato, $p$-adic étale cohomology , Publ. Math. Inst. Hautes Étud. Sci., 63 (1986), 107–152
  • M B ökstedtW-C Hsiang I Madsen, The cyclotomic trace and algebraic $K$-theory of spaces , Invent. Math., 111 (1993), 865–940
  • M B ökstedt I Madsen, Topological cyclic homology of the integers , Asterisque, 226 (1994), 57–143
  • M B ökstedt I Madsen, Algebraic $K$-theory of local number fields: the unramified case , Prospects in topology, Princeton, NJ, 1994, Ann. of Math. Studies, 138, , Princeton University Press (1995), 28–57
  • A K Bousfield, The localization of spectra with respect to homology , Topology, 18 (1979), 257–281
  • B I Dundas, Relative $K$-theory and topological cyclic homology , Acta Math., 179 (1997), 223–242
  • W G Dwyer, Twisted homological stability for general linear groups , Ann. of Math., 111 (1980), 239–251
  • W G Dwyer E M Friedlander, Algebraic and étale $K$-theory , Trans. AMS, 292 (1985), 247–280
  • W G Dwyer E M Friedlander, Topological models for arithmetic , Topology, 33 (1994), 1–24
  • W G Dwyer S A Mitchell, On the $K$-theory spectrum of a ring of algebraic integers , $K$-Theory, 14 (1998), 201–263
  • F T Farrell W C Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds , Algebr. geom. Topol. Stanford/Calif. 1976, Proc. Symp. Pure Math., 32, Part 1 (1978), 325–337
  • L Hesselholt I Madsen, On the $K$-theory of finite algebras over Witt vectors of perfect fields , Topology, 36 (1997), 29–101
  • R T Hoobler, When is $\Br(X)=\Br'(X)$? , Brauer groups in ring theory and algebraic geometry, Proc. Antwerp 1981, Lecture Notes in Math., 917 , Springer-Verlag (1982), 231–244
  • K Igusa, The stability theorem for smooth pseudoisotopies , $K$-Theory, 2 (1988), 1–355
  • J.R Klein J Rognes, The fiber of the linearization map $A(*) \to K(\Z)$ , Topology, 36 (1997), 829–848
  • K-H Knapp, $Im(J)$-theory for torsion-free spaces The complex projective space as an example , Revised version of Habilitationsschrift Bonn 1979, in preparation
  • S Lichtenbaum, On the values of zeta and $L$-functions, I , Annals of Math., 96 (1972), 338–360 \ref
  • \key Li2 S Lichtenbaum, Values of zeta functions, étale cohomology, and algebraic $K$-theory , Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic, Lecture Notes in Math., 342 , Springer-Verlag (1973), 489–501
  • I Madsen R J Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, 92, , Princeton University Press (1979)
  • I Madsen C Schlichtkrull, The circle transfer and $K$-theory (Grove, Karsten et al, Geometry and topology, Aarhus Proceedings of the conference on geometry and topology, Aarhus, Denmark, August 10–16, 1998, eds.), Contemp. Math., 258 , American Mathematical Society (2000), 307–328
  • I MadsenV Snaith J Tornehave, Infinite loop maps in geometric topology , Math. Proc. Camb. Phil. Soc., 81 (1977), 399–429
  • I Madsen U Tillmann, The stable mapping class group and $Q(\C P^\infty_+)$ , Invent. Math., 145 (2001), 509–544
  • J P MayF Quinn N Ray with contributions by J Tornehave, $E_\infty$ ring spaces and $E_\infty$ ring spectra, Lecture Notes in Math., 577 , Springer-Verlag (1977)
  • J S Milne, Étale cohomology, Princeton Mathematical Series, 33, , Princeton University Press (1980)
  • J S Milne, Arithmetic duality theorems, Perspectives in Mathematics, 1 , Academic Press, Inc. (1986)
  • G Mislin, Localization with respect to $K$-theory , J. Pure Appl. Algebra, 10 (1977/78), 201–213
  • S A Mitchell, On $p$-adic topological $K$-theory (P G Goerss et al, Algebraic $K$-theory and algebraic topology, Proceedings of the NATO Advanced Study Institute, Lake Louise, Alberta, Canada, December 12–16, 1991, eds.), NATO ASI Ser. Ser. C, Math. Phys. Sci., 407 , Kluwer Academic Publishers (1993), 197–204
  • R E Mosher, Some stable homotopy of complex projective space , Topology, 7 (1968), 179–193
  • J Mukai, The $S^1$-transfer map and homotopy groups of suspended complex projective spaces , Math. J. Okayama Univ., 24 (1982), 179–200
  • D Quillen, Higher algebraic $K$-theory. I , Algebraic $K$-Theory, I: Higher $K$-theories, Lecture Notes Math., 341 , Springer-Verlag (1973), 85–147
  • D Quillen, Finite generation of the groups $K_i$ of rings of algebraic integers , Algebraic $K$-Theory, I: Higher $K$-theories, Lecture Notes in Math., 341 , Springer-Verlag (1973), 179–198
  • D Quillen, Higher algebraic $K$-theory , Proc. Intern. Congress Math. Vancouver, 1974, I , Canad. Math. Soc. (1975), 171–176
  • D Quillen, Letter from Quillen to Milnor on $\operatorname{Im}(\pi_i O \to \pi_i^s \to K_i\Z)$ , Algebraic $K$-theory, Proc. Conf. Northwestern Univ. Evanston, Ill. 1976, Lecture Notes in Math., 551 , Springer-Verlag (1976), 182–188
  • D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Math., 121 , Academic Press (1986)
  • J Rognes, Two-primary algebraic $K$-theory of pointed spaces , Topology, 41 (2002), 873–926
  • J Rognes C Weibel, Two–primary algebraic $K$-theory of rings of integers in number fields , J. Am. Math. Soc., 13 (2000), 1–54
  • N E Steenrod, Cohomology operations, Annals of Mathematics Studies, 50, , Princeton University Press (1962)
  • A Suslin V Voevodsky, Bloch–Kato conjecture and motivic cohomology with finite coefficients (B B Gordon et al, The arithmetic and geometry of algebraic cycles, eds.), NATO ASI Ser. Ser. C, Math. Phys. Sci., 548 , Kluwer Academic Publishers (2000), 117–189
  • J Tate, Duality theorems in Galois cohomology over number fields , Proc. Intern. Congress Math. Stockholm, 1962, Inst. Mittag–Leffler (1963), 234–241
  • H Toda, A topological proof of theorems of Bott and Borel–Hirzebruch for homotopy groups of unitary groups , Mem. Coll. Sci. Univ. Kyoto, Ser. A, 32 (1959), 103–119
  • V Voevodsky, The Milnor Conjecture , preprint (1996)
  • V Voevodsky, On $2$-torsion in motivic cohomology , preprint (2001)
  • F Waldhausen, Algebraic $K$-theory of topological spaces. I , Algebr. geom. Topol. Stanford/Calif. 1976, Proc. Symp. Pure Math., 32, Part 1 (1978), 35–60
  • F Waldhausen, Algebraic $K$-theory of topological spaces, II , Algebraic topology, Proc. Symp. Aarhus 1978, Lecture Notes in Math., 763 , Springer-Verlag (1979), 356–394
  • 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), 141–184
  • F Waldhausen, Algebraic $K$-theory of spaces , Algebraic and geometric topology, Proc. Conf. New Brunswick/USA 1983, Lecture Notes in Math., 1126 , Springer-Verlag (1985), 318–419
  • F Waldhausen, Algebraic $K$-theory of spaces, concordance, and stable homotopy theory , Algebraic topology and algebraic $K$-theory, Proc. Conf. Princeton, NJ (USA), Ann. Math. Stud., 113 (1987), 392–417
  • F Waldhausen et al, The stable parametrized $h$-cobordism theorem , in preparation