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)

 
 

 

On $ H\sb *(\Omega\sp {n+2}S\sp {n+1};{\bf F}\sb 2)$


Author: Thomas J. Hunter
Journal: Trans. Amer. Math. Soc. 314 (1989), 405-420
MSC: Primary 55P35; Secondary 55S12, 55T20
DOI: https://doi.org/10.1090/S0002-9947-1989-0946423-3
MathSciNet review: 946423
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we study $ {H_\ast}{\Omega ^{n + 2}}{S^{n + 1}}$ . Here $ \Omega X$ denotes the space of pointed maps $ {S^1} \to X$, and $ {H_\ast}$ represents homology modulo $ 2$. We show that the Eilenberg-Moore spectral sequence $ \operatorname{Tor}_{{H^\ast }\Omega _0^{n + 1}{S^{n + 1}}}^{\ast\ast}({F_{2,}}{F_2}) \Rightarrow {H^\ast }{\Omega ^{n + 2}}{S^{n + 1}}$ collapses, and we identify the kernel of the Whitehead product map $ {\Omega ^{n + 1}}{p_\ast}:{H_\ast}{\Omega ^{n + 3}}{S^{2n + 1}} \to {H_\ast}{\Omega ^{n + 1}}{S^n}$ . These observations yield two different descriptions of $ {H_\ast}{\Omega ^{n + 2}}{S^{n + 1}}$ up to extension.


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

  • [1] T. Kudo and S. Araki, On $ {H_\ast}({\Omega ^N}{S^n};{Z_2})$, Proc. Japan Acad. 32 (1956), 333-335. MR 0079768 (18:143d)
  • [2] W. Browder, The cohomology of covering space of $ H$-spaces, Bull. Amer. Math. Soc. 65 (1959), 140-141. MR 0111024 (22:1891)
  • [3] A. Clark, Homotopy-commutativity and the Moore spectral sequence, Pacific J. Math. 15 (1965), 65-74. MR 0177416 (31:1679)
  • [4] F. R. Cohen, T. J. Lada and J. P. May, The homology of iterated loop spaces, Lecture Notes in Math., vol. 533, Springer, 1976. MR 0436146 (55:9096)
  • [5] F. R. Cohen and F. P. Peterson, Mimeographed handwritten notes.
  • [6] W. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1974), 255-265. MR 0394663 (52:15464)
  • [7] J. W. Milnor and J. C. Moore, On the structure of Hopf-algebras, Ann. of Math. 81 (1965), 211-264. MR 0174052 (30:4259)
  • [8] H. Toda, Composition methods in the homotopy groups of spheres, Princeton Univ. Press, 1962. MR 0143217 (26:777)
  • [9] Robert J. Wellington, The unstable Adams spectral sequence for free iterated loop spaces, Mem. Amer. Math. Soc. 36, no. 258, 1982. MR 646741 (83c:55028)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55P35, 55S12, 55T20

Retrieve articles in all journals with MSC: 55P35, 55S12, 55T20


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0946423-3
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society