Local 𝐻-maps of 𝐵𝑈 and applications to smoothing theory

Lance, Timothy

doi:10.1090/S0002-9947-1988-0957078-5

Local $H$-maps of $B\textrm {U}$ and applications to smoothing theory
HTML articles powered by AMS MathViewer

by Timothy Lance

Trans. Amer. Math. Soc. 309 (1988), 391-424

DOI: https://doi.org/10.1090/S0002-9947-1988-0957078-5

PDF | Request permission

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

J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632. MR 139178, DOI 10.2307/1970213
J. F. Adams, On the groups $J(X)$. II, Topology 3 (1965), 137–171. MR 198468, DOI 10.1016/0040-9383(65)90040-6
J. F. Adams, On the groups $J(X)$. IV, Topology 5 (1966), 21–71. MR 198470, DOI 10.1016/0040-9383(66)90004-8

Lectures on generalized cohomology

Stable homotopy and generalized homology

J. F. Adams and S. B. Priddy, Uniqueness of $B\textrm {SO}$, Math. Proc. Cambridge Philos. Soc. 80 (1976), no. 3, 475–509. MR 431152, DOI 10.1017/S0305004100053111
M. F. Atiyah and G. B. Segal, Exponential isomorphisms for $\lambda$-rings, Quart. J. Math. Oxford Ser. (2) 22 (1971), 371–378. MR 291250, DOI 10.1093/qmath/22.3.371

On the Kummer congruences and the stable homotopy of

C. Berge, Principles of combinatorics, Mathematics in Science and Engineering, Vol. 72, Academic Press, New York-London, 1971. Translated from the French. MR 0270922
A. Borel and J.-P. Serre, Groupes de Lie et puissances réduites de Steenrod, Amer. J. Math. 75 (1953), 409–448 (French). MR 58213, DOI 10.2307/2372495
Raoul Bott, Lectures on $K(X)$, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0258020
R. Bott and J. Milnor, On the parallelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87–89. MR 102804, DOI 10.1090/S0002-9904-1958-10166-4
William Browder, Torsion in $H$-spaces, Ann. of Math. (2) 74 (1961), 24–51. MR 124891, DOI 10.2307/1970305
William Browder, Higher torsion in $H$-spaces, Trans. Amer. Math. Soc. 108 (1963), 353–375. MR 155326, DOI 10.1090/S0002-9947-1963-0155326-8
G. Brumfiel, On the homotopy groups of $\textrm {BPL}$ and $\textrm {PL/O}$, Ann. of Math. (2) 88 (1968), 291–311. MR 234458, DOI 10.2307/1970576

Differentiable

actions on homotopy spheres

G. Brumfiel, On integral $\textrm {PL}$ characteristic classes, Topology 8 (1969), 39–46. MR 234484, DOI 10.1016/0040-9383(69)90029-9
G. Brumfiel, Homotopy equivalences of almost smooth manifolds, Comment. Math. Helv. 46 (1971), 381–407. MR 305419, DOI 10.1007/BF02566851
Francis Clarke, Self-maps of $B\textrm {U}$, Math. Proc. Cambridge Philos. Soc. 89 (1981), no. 3, 491–500. MR 602302, DOI 10.1017/S0305004100058382
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 0436146, DOI 10.1007/BFb0080464
Samuel Eilenberg and John C. Moore, Homology and fibrations. I. Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966), 199–236. MR 203730, DOI 10.1007/BF02564371
Bruce Ferrero and Lawrence C. Washington, The Iwasawa invariant $\mu _{p}$ vanishes for abelian number fields, Ann. of Math. (2) 109 (1979), no. 2, 377–395. MR 528968, DOI 10.2307/1971116
V. K. A. M. Gugenheim and J. Peter May, On the theory and applications of differential torsion products, Memoirs of the American Mathematical Society, No. 142, American Mathematical Society, Providence, R.I., 1974. MR 0394720
Morris W. Hirsch and Barry Mazur, Smoothings of piecewise linear manifolds, Annals of Mathematics Studies, No. 80, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0415630
Dale Husemoller, The structure of the Hopf algebra $H_\ast (BU)$ over a Z$_{(p)}$-algebra, Amer. J. Math. 93 (1971), 329–349. MR 286867, DOI 10.2307/2373380

On the homology of the fiber of

Dale Husemoller, John C. Moore, and James Stasheff, Differential homological algebra and homogeneous spaces, J. Pure Appl. Algebra 5 (1974), 113–185. MR 365571, DOI 10.1016/0022-4049(74)90045-0
Kenkichi Iwasawa, On some invariants of cyclotomic fields, Amer. J. Math. 80 (1958), 773–783; erratum: 81 (1958), 280. MR 0124317, DOI 10.2307/2372857
Donald W. Kahn, Induced maps for Postnikov systems, Trans. Amer. Math. Soc. 107 (1963), 432–450. MR 150777, DOI 10.1090/S0002-9947-1963-0150777-X
Donald W. Kahn, Differentiable approximations to homotopy resolutions and framed cobordism, Pacific J. Math. 113 (1984), no. 2, 373–382. MR 749542, DOI 10.2140/pjm.1984.113.373
Timothy Lance, Local $H$-maps of classifying spaces, Trans. Amer. Math. Soc. 254 (1979), 195–215. MR 539915, DOI 10.1090/S0002-9947-1979-0539915-6
Timothy Lance, Steenrod and Dyer-Lashof operations on $B\textbf {U}$, Trans. Amer. Math. Soc. 276 (1983), no. 2, 497–510. MR 688957, DOI 10.1090/S0002-9947-1983-0688957-X
R. Lashof and M. Rothenberg, Microbundles and smoothing, Topology 3 (1965), 357–388. MR 176480, DOI 10.1016/0040-9383(65)90003-0
Arunas Liulevicius, On characteristic classes, Aarhus Universitet, Matematisk Institut, Aarhus, 1968. Lectures given at the Nordic Summer School in Mathematics (June 16th-July 6th, 1968). MR 0256409
Saunders MacLane, Homology, 1st ed., Die Grundlehren der mathematischen Wissenschaften, Band 114, Springer-Verlag, Berlin-New York, 1967. MR 0349792
Ib Madsen and R. James Milgram, The classifying spaces for surgery and cobordism of manifolds, Annals of Mathematics Studies, No. 92, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. MR 548575
I. Madsen, V. Snaith, and J. Tornehave, Infinite loop maps in geometric topology, Math. Proc. Cambridge Philos. Soc. 81 (1977), no. 3, 399–430. MR 494076, DOI 10.1017/S0305004100053482
J. Peter May, $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. MR 0494077
J. Milnor, On the cobordism ring $\Omega ^{\ast }$ and a complex analogue. I, Amer. J. Math. 82 (1960), 505–521. MR 119209, DOI 10.2307/2372970
J. Milnor, On characteristic classes for spherical fibre spaces, Comment. Math. Helv. 43 (1968), 51–77. MR 226638, DOI 10.1007/BF02564380
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554, DOI 10.1515/9781400881826
Robert E. Mosher and Martin C. Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, Publishers, New York-London, 1968. MR 0226634
Hans J. Munkholm, The Eilenberg-Moore spectral sequence and strongly homotopy multiplicative maps, J. Pure Appl. Algebra 5 (1974), 1–50. MR 350735, DOI 10.1016/0022-4049(74)90002-4
Erna H. Pearson, On the congruences $(p-1)!\equiv -1$ and $2^{p-1}\equiv 1\,(\textrm {mod}\,p^{2})$, Math. Comp. 17 (1964), 194–195. MR 159780, DOI 10.1090/S0025-5718-1963-0159780-0
Franklin P. Peterson, The $\textrm {mod}\,p$ homotopy type of $\textrm {BSO}$ and $F/\textrm {PL}$, Bol. Soc. Mat. Mexicana (2) 14 (1969), 22–27. MR 259911
Daniel Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR 279804, DOI 10.1016/0040-9383(71)90018-8
William M. Singer, Connective fiberings over $\textrm {BU}$ and $\textrm {U}$, Topology 7 (1968), 271–303. MR 232392, DOI 10.1016/0040-9383(68)90006-2
Victor Snaith, The complex $J$-homomorphism. I, Proc. London Math. Soc. (3) 34 (1977), no. 2, 269–302. MR 442936, DOI 10.1112/plms/s3-34.2.269
Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554, DOI 10.1007/978-1-4684-9322-1

The image of

as an

space

James Stasheff, More characteristic classes for spherical fibre spaces, Comment. Math. Helv. 43 (1968), 78–86. MR 226639, DOI 10.1007/BF02564381
Dennis Sullivan, Geometric topology. Part I, Massachusetts Institute of Technology, Cambridge, Mass., 1971. Localization, periodicity, and Galois symmetry; Revised version. MR 0494074
Dennis Sullivan, Genetics of homotopy theory and the Adams conjecture, Ann. of Math. (2) 100 (1974), 1–79. MR 442930, DOI 10.2307/1970841
Hirosi Toda, $p$-primary components of homotopy groups. I. Exact sequences in Steenrod algebra, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 31 (1958), 129–142. MR 105682, DOI 10.1215/kjm/1250776909
Akihiro Tsuchiya, Characteristic classes for PL micro bundles, Nagoya Math. J. 43 (1971), 169–198. MR 314062, DOI 10.1017/S002776300001446X
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0