Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On compositions of the loop and suspension functors

Author: M. H. Eggar
Journal: Proc. Amer. Math. Soc. 123 (1995), 597-606
MSC: Primary 55P65; Secondary 55P62
MathSciNet review: 1218114
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem studied is whether, from knowledge of the homotopy type of $ {\Omega ^{{d_k}}}{\Sigma ^{{c_k}}} \cdots {\Omega ^{{d_2}}}{\Sigma ^{{c_2}}}{\Omega ^{{d_1}}}{\Sigma ^{{c_1}}}X = MX$ for suitable spaces X, one can recover the nonnegative integers $ {c_1},{d_1}, \ldots ,{c_k},{d_k}$. The Betti numbers of X and $ {c_1},{d_1}, \ldots ,{c_k},{d_k}$ do determine the ith Betti number of MX, but even for small k , i and for X a sphere (say) the answer is a complicated one, since it depends on parities and graded Witt numbers depending on graded Witt numbers. It is shown that k can be found and that $ {c_i},{d_j}$ can always be determined up to finitely many possibilities and usually uniquely.

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

  • [1] D. J. Anick, The computation of rational homotopy groups is #P-hard, Computers in geometry and topology (Chicago IL, 1986), Lecture Notes in Pure and Appl. Math., vol. 114, Dekker, New York, 1989, pp. 1-56. MR 988689 (89m:55006)
  • [2] I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170-197. MR 0073181 (17:396b)
  • [3] T. Miller and J. Neisendorfer, Formal and coformal spaces, Illinois J. of Math. 22 (1978), 565-580. MR 0500938 (58:18429)
  • [4] E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937), 152-160.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 55P65, 55P62

Retrieve articles in all journals with MSC: 55P65, 55P62

Additional Information

Keywords: Loop, suspension, homotopy equivalence, rational homotopy theory
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society