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)



Rational Moore $ G$-spaces

Author: Peter J. Kahn
Journal: Trans. Amer. Math. Soc. 298 (1986), 245-271
MSC: Primary 55N25; Secondary 55P62, 55Q05, 57S17
MathSciNet review: 857443
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper obtains some existence and uniqueness results for Moore spaces in the context of the equivariant homotopy theory of Bredon. This theory incorporates fixed-point-set data as part of the structure and so is a refinement of the classical equivariant homotopy theory. To avoid counterexamples to existence in the classical case and to focus on new phenomena involving the fixed-point-set structure, most of the results involve rational spaces. In this setting, there are no obstacles to existence, but a notion of projective dimension presents an obstacle to uniqueness: uniqueness is proved, subject to constraint on the projective dimension, and an example shows that this constraint is sharp. Various related existence results are proved and computations are given of certain equivariant mapping sets $ [X,\,Y]$, $ X$ an equivariant Moore space.

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

  • [A] James E. Arnold, On Steenrood's problem for cyclic $ p$-groups, Canad. J. Math. 29 (1977), 421-428. MR 0440536 (55:13411)
  • [B] G. Bredon, Equivariant cohomology theories, Lecture Notes in Math., vol. 34, Springer-Verlag, New York, 1967. MR 0214062 (35:4914)
  • [C1] G. Carlsson, A counterexample to a conjecture of Steenrod, Invent. Math. 64 (1981), 171-174. MR 621775 (82j:57036)
  • [C2] -, An Adams-type spectral sequence for change of rings, Houston J. Math. 4 (1978), 541-550. MR 523612 (80c:18009)
  • [Co] G. Gooke, Replacing homotopy actions by topological actions, Trans. Amer. Math. Soc. 237 (1978), 391-406. MR 0461544 (57:1529)
  • [E] A. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983), 275-284. MR 690052 (84f:57029)
  • [H] P. J. Hilton, Homotopy theory and duality, Gordon & Breach, New York, 1965. MR 0198466 (33:6624)
  • [K1] P. J. Kahn, Steenrod's problem and $ k$-invariants of certain classifying spaces, Algebraic $ K$-theory. Part II, Lecture Notes in Math., vol. 967, Springer-Verlag, New York, 1982. MR 689393 (84k:57030)
  • [K2] -, Equivariant homology decompositions, Trans. Amer. Math. Soc. 298 (1986), 273-287. MR 857444 (87j:55010)
  • [L] W. Lück, Diplomarbeit, Univ. of Göttingen, 1981.
  • [M] S. Mac Lane, Homology, Academic Press and Springer-Verlag, New York, 1963. MR 1344215 (96d:18001)
  • [R & T] M. Rothenberg and G. Triantafillou, An algebraic model for simple homotopy types, Univ. of Minnesota Math. Report 83-112, 1984.
  • [S] J. Smith, Equivariant Moore spaces, 1982 (preprint). MR 802793 (87e:55023)
  • [Sw] R. Swan, Invariant rational functions and a problem of Steenrod, Invent. Math. 7 (1969), 148-158. MR 0244215 (39:5532)
  • [T1] G. Triantafillou, Equivariant minimal models, Trans. Amer. Math. Soc. 274 (1982), 509-532. MR 675066 (84g:55017)
  • [T2] -, $ G$-spaces with prescribed equivariant cohomology, Univ. of Minnesota (preprint).
  • [T3] -, Rationalization of Hopf $ G$-spaces, Math. Z. 182 (1983), 485-500. MR 701365 (84h:55008)
  • [V] P. Vogel, On Steenrod's problem for non-abelian finite groups, Algebraic Topology, Aarhus 1982, Lecture Notes in Math., vol. 1051, Springer-Verlag, New York, 1984. MR 764607 (86e:55015)
  • [W] C. T. C. Wall, Finiteness conditions for CW complexes. II, Proc. Roy. Soc. A 295 (1966), 129-139. MR 0211402 (35:2283)
  • [Wn] S. Waner, Equivariant homotopy theory and Milnor's theorem, Trans. Amer. Math. Soc. 258 (1980), 351-368. MR 558178 (82m:55016a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55N25, 55P62, 55Q05, 57S17

Retrieve articles in all journals with MSC: 55N25, 55P62, 55Q05, 57S17

Additional Information

Keywords: Moore space, equivariant, $ G$-space, homotopy
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society