Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Local $ H$-maps of $ B{\rm U}$ and applications to smoothing theory


Author: Timothy Lance
Journal: Trans. Amer. Math. Soc. 309 (1988), 391-424
MSC: Primary 55P47; Secondary 55N15, 57R10
DOI: https://doi.org/10.1090/S0002-9947-1988-0957078-5
MathSciNet review: 957078
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: When localized at an odd prime $ p$, the classifying space $ PL/O$ for smoothing theory splits as an infinite loop space into the product $ C \times N$ where $ C = {\text{Cokernel}}\,(J)$ and $ N$ is the fiber of a $ p$-local $ H$-map $ BU \to BU$. This paper studies spaces which arise in this latter fashion, computing the cohomology of their Postnikov towers and relating their $ k$-invariants to properties of the defining self-maps of $ BU$. If $ Y$ is a smooth manifold, the set of homotopy classes $ [Y,\,N]$ is a certain subgroup of resmoothings of $ Y$, and the $ k$-invariants of $ N$ generate obstructions to computing that subgroup. These obstructions can be directly related to the geometry of $ Y$ and frequently vanish.


References [Enhancements On Off] (What's this?)

  • [1] J. F. Adams, Vector fields on spheres, Ann. of Math. 75 (1962), 603-632. MR 0139178 (25:2614)
  • [2] -, On the groups $ J(X)$. II, Topology 3 (1965), 137-171. MR 0198468 (33:6626)
  • [3] -, On the groups $ J(X)$. IV, Topology 5 (1966), 21-71. MR 0198470 (33:6628)
  • [4] -, Lectures on generalized cohomology, Category Theory, Homology Theory, and Applications, III, Lecture Notes in Math., vol. 99, Springer-Verlag, Berlin and New York, 1969.
  • [5] -, Stable homotopy and generalized homology, Univ. of Chicago Press, 1974.
  • [6] J. F. Adams and S. Priddy, Uniqueness of $ BSO$, Proc. Cambridge Philos. Soc. 80 (1976), 475-509. MR 0431152 (55:4154)
  • [7] M. F. Atiyah and G. B. Segal, Exponential isomorphisms for $ \lambda $-rings, Quart. J. Math. Oxford (2) 22 (1971), 371-378. MR 0291250 (45:344)
  • [8] A. Baker, F. Clarke, N. Ray, and L. Schwartz, On the Kummer congruences and the stable homotopy of $ BU$, preprint, 1986.
  • [9] C. Berge, Principles of combinatorics, Academic Press, New York, London, 1971. MR 0270922 (42:5805)
  • [10] A Borel and J. P. Serre, Groupes de Lie et puissances reduites de Steenrod, Amer. J. Math. 75 (1953), 409-448. MR 0058213 (15:338b)
  • [11] R. Bott, Lectures on $ K(X)$, W. A. Benjamin, New York, 1969. MR 0258020 (41:2667)
  • [12] R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87-89. MR 0102804 (21:1590)
  • [13] W. Browder, Torsion in $ H$-spaces, Ann. of Math. (2) 74 (1961), 24-51. MR 0124891 (23:A2201)
  • [14] -, Higher torsion in $ H$-spaces, Trans. Amer. Math. Soc. 108 (1963), 353-375. MR 0155326 (27:5260)
  • [15] G. Brumfiel, On the homotopy groups of $ BPL$ and $ PL/O$, Ann. of Math. (2) 88 (1968), 291-311. MR 0234458 (38:2775)
  • [16] -, Differentiable $ {S^1}$ actions on homotopy spheres, mimeo. notes, 1969.
  • [17] -, On integral $ PL$ characteristic classes, Topology 8 (1969), 39-46. MR 0234484 (38:2801)
  • [18] -, Homotopy equivalences of almost smooth manifolds, Comment. Math. Helv. 46 (1971), 381-407. MR 0305419 (46:4549)
  • [19] F. Clarke, Self maps of $ BU$, Proc. Cambridge Philos. Soc. 89 (1981), 491-500. MR 602302 (82f:55007)
  • [20] F. Cohen, T. Lada, and J. P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol 533, Springer-Verlag, Berlin and New York, 1976. MR 0436146 (55:9096)
  • [21] S. Eilenberg and J. C. Moore, Homology and fibrations. I, Comment. Math. Helv. 40 (1966), 199-236. MR 0203730 (34:3579)
  • [22] B. Ferrero and L. Washington, The Iwasawa invariant $ {\mu _p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), 377-395. MR 528968 (81a:12005)
  • [23] V. K. A. M. Gugenheim and J. P. May, On the theory and applications of differential torsion products, Mem. Amer. Math. Soc. No. 142, 1974. MR 0394720 (52:15519)
  • [24] M. Hirsch and B. Mazur, Smoothings of differentiable manifolds, Ann. of Math. Studies, No. 80, Princeton Univ. Press, Princeton, N.J., 1974. MR 0415630 (54:3711)
  • [25] D. Husemoller, The structure of the Hopf algebra $ {H_{\ast}}(BU)$ over a $ {Z_{(p)}}$ algebra, Amer. J. Math. 43 (1971), 329-349. MR 0286867 (44:4074)
  • [26] -, On the homology of the fiber of $ {\psi ^q} - 1$, Algebraic $ K$-Theory. I, Lecture Notes in Math., Springer-Verlag, Berlin and New York, 1973, pp. 199-204.
  • [27] D. Husemoller, J. C. Moore, and J. Stasheff, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113-185. MR 0365571 (51:1823)
  • [28] K. Iwasaw, On some invariants of cyclotomic fields, Amer. J. Math. 80 (1958), 773-783; erratum 81 (1959), 280. MR 0124317 (23:A1631)
  • [29] D. W. Kahn, Induced maps for Postnikov systems, Trans. Amer. Math. Soc. 107 (1963), 432-450. MR 0150777 (27:764)
  • [30] -, Differential approximations to homotopy resolutions and framed corbodism, Pacific J. Math. 113 (1984), 373-382. MR 749542 (86d:57017)
  • [31] T. Lance, Local $ H$-maps of classifying spaces, Trans. Amer. Math. Soc. 254 (1979), 195-215. MR 539915 (80k:55051)
  • [32] -, Steenrod and Dyer-Lashof operations on $ BU$, Trans. Amer. Math. Soc. 276 (1983), 497-510. MR 688957 (85d:55027)
  • [33] R. Lashof and M. Rothenberg, Microbundles and smoothing, Topology 3 (1965), 357-388. MR 0176480 (31:752)
  • [34] A. Liulevicius, On characteristic classes, Nordic Summer School Notes, Aarhus Univ., 1968. MR 0256409 (41:1065)
  • [35] S. MacLane, Homology, Springer-Verlag, Berlin and New York, 1967. MR 0349792 (50:2285)
  • [36] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Ann. of Math. Studies, No. 92, Princeton Univ. Press, Princeton, N.J., 1979. MR 548575 (81b:57014)
  • [37] I. Madsen, V. Snaith, and J. Tornehave, Infinite loop maps in geometric topology, Math. Proc. Cambridge Philos. Soc. 81 (1977), 399-430. MR 0494076 (58:13007)
  • [38] J. P. May (with contributions by F. Quinn, N. Ray, and J. Tornhave), $ {E_\infty }$ ring spaces and $ {E_\infty }$ ring spectra, Lecture Notes in Math., vol. 577, Springer-Verlag, Berlin and New York, 1977. MR 0494077 (58:13008)
  • [39] J. Milnor, On the cobordism ring $ {\Omega _{\ast}}$ and a complex analogue. I, Amer. J. Math. 82 (1960), 505-521. MR 0119209 (22:9975)
  • [40] -, On characteristic classes for spherical fiber spaces, Comment. Math. Helv. 43 (1968), 51-77. MR 0226638 (37:2227)
  • [41] J. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 0174052 (30:4259)
  • [42] J. Milnor and J. Stasheff, Characteristic classes, Ann. of Math. Studies, No. 76, Princeton Univ. Press, Princeton, N. J., 1974. MR 0440554 (55:13428)
  • [43] R. Mosher and M. Tangora, Cohomology operations and applications in homotopy theory, Harper and Row, New York, 1968. MR 0226634 (37:2223)
  • [44] H. Munkholm, The Eilenberg-Moore spectral sequences and strongly homotopy multiplicative maps, J. Pure Appl. Algebra 5 (1974), 1-50. MR 0350735 (50:3227)
  • [45] E. H. Pearson, On the congruences $ (p - 1)! \equiv - 1$ and $ {2^{p - 1}} \equiv 1\,\bmod \,{p^2}$, Math. Comp. 17 (1963), 194-195. MR 0159780 (28:2996)
  • [46] F. P. Peterson, The $ \bmod \,p$ homotopy type of $ BSO$ and $ F/PL$, Bol. Soc. Math. Mexicana 14 (1969), 22-27. MR 0259911 (41:4540)
  • [47] D. Quillen, The Adams conjecture, Topology 10 (1971), 67-80. MR 0279804 (43:5525)
  • [48] W. Singer, Connective fibrings over $ BU$ and $ U$, Topology 7 (1968), 271-303. MR 0232392 (38:717)
  • [49] V. Snaith, The complex $ J$-homomorphism, Proc. London Math. Soc. 34 (1977), 269-302. MR 0442936 (56:1311)
  • [50] E. Spanier, Algebraic topology, Springer-Verlag, Berlin and New York, 1966. MR 666554 (83i:55001)
  • [51] J. Stasheff, The image of $ J$ as an $ H$-space $ \bmod \,p$, Conf. on Algebraic Topology, Univ. of Illinois at Chicago Circle, 1968.
  • [52] -, More characteristic classes of spherical fiber spaces, Comment. Math. Helv. 43 (1968), 78-86. MR 0226639 (37:2228)
  • [53] D. Sullivan, Geometric topology. Part I, Localization, periodicity, and Galois symmetry, mimeo. notes, M.I.T., 1970. MR 0494074 (58:13006a)
  • [54] -, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. 100 (1974), 1-79. MR 0442930 (56:1305)
  • [55] H. Toda, $ p$ primary components of homotopy groups. I, Mem. Coll. Sci. Kyoto Univ. Ser. A Math. 31 (1959), 129-142. MR 0105682 (21:4420a)
  • [56] A. Tsuchiyah, Characteristic classes for $ PL$ microbundles, Nagoya Math. J. 43 (1971), 169-198. MR 0314062 (47:2614)
  • [57] G. W. Whitehead, Elements of homotopy theory, Springer-Verlag, Berlin and New York, 1978. MR 516508 (80b:55001)
  • [58] J. Wolf, Ph.D. thesis, Brown University, 1973.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55P47, 55N15, 57R10

Retrieve articles in all journals with MSC: 55P47, 55N15, 57R10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0957078-5
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society