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)



Hybrid spaces with interesting cohomology

Author: Kathryn Lesh
Journal: Trans. Amer. Math. Soc. 347 (1995), 3247-3262
MSC: Primary 55S10; Secondary 55P15, 55P60
MathSciNet review: 1316857
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ p$ be an odd prime, and let $ R$ be a polynomial algebra over the Steenrod algebra with generators in dimensions prime to $ p$. To such an algebra is associated a $ p$-adic pseudoreflection group $ W$, and we assume that $ W$ is of order prime to $ p$ and irreducible. Adjoin to $ R$ a one-dimensional element $ z$, and give $ R[z]$ an action of the Steenrod algebra by $ \beta z = 0$ and $ \beta x = (\left\vert x \right\vert/2)zx$ for an even dimensional element $ x$. We show that the subalgebra of elements of $ R[z]$ consisting of elements of degree greater than one is realized uniquely, up to homotopy, as the cohomology of a $ p$-complete space. This space can be thought of as a cross between spaces studied by Aguade, Broto, and Notbohm, and the Clark-Ewing examples, further studied by Dwyer, Miller, and Wilkerson.

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

  • [ABN] J. Aguade, C. Broto, and D. Notbohm, Homotopy classification of spaces with interesting cohomology and a conjecture of Cooke. I, Topology 33 (1994) 455-492. MR 1286926 (95i:55006)
  • [B-K] A. Bousfield and D. Kan, Homotopy limits, completions, and localizations, Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin and New York, 1972. MR 0365573 (51:1825)
  • [Brown] K. Brown, Cohomology of groups, Graduate Texts in Math., vol. 87, Springer-Verlag, New York, 1982. MR 672956 (83k:20002)
  • [C-E] A. Clark and J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434. MR 0367979 (51:4221)
  • [DF-S] E. Dror Farjoun and J. Smith, A geometric interpretation of Lannes' functor $ T$, Astérisque 191 (1990), 87-95. MR 1098968 (92h:55013)
  • [Dwyer] W. Dwyer, Strong convergence of the Eilenberg-Moore spectral sequence, Topology 13 (1974), 255-265. MR 0394663 (52:15464)
  • [DMW] W. Dwyer, H. Miller, and C. Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 29-45. MR 1153237 (92m:55013)
  • [D-W] W. Dwyer and C. Wilkerson, Spaces of null homotopic maps, Proceedings of the 1988 Luminy Conference, Astérisque 191 (1990), 97-108. MR 1098969 (92b:55004)
  • [D-Z] W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Proceedings of the 1986 Barcelona Conference on Algebraic Topology, J. Aguade, and R. Kane, eds., Lecture Notes in Math., vol. 1298, Springer-Verlag, Berlin and New York, 1988. MR 928826 (89b:55018)
  • [Lannes] J. Lannes, Sur les espaces fonctionels dont la source est le classifiant d'un $ p$-groupe abelièn elémentaire, Publ. Math. Inst. Hautes Etudes Sci. 75 (1992), 135-244. MR 1179079 (93j:55019)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 55S10, 55P15, 55P60

Retrieve articles in all journals with MSC: 55S10, 55P15, 55P60

Additional Information

Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society