Co-H-structures on equivariant Moore spaces

Martin Arkowitz; Marek Golasiński

Fundamenta Mathematicae (1994)

  • Volume: 146, Issue: 1, page 59-67
  • ISSN: 0016-2736

Abstract

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Let G be a finite group, 𝕆 G the category of canonical orbits of G and A : 𝕆 G 𝔸 b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with E x t n - 1 ( A , A A ) . Then the case G = p k leads to an example of infinitely many G-homotopically distinct G-maps φ i : X Y such that φ i H , φ j H : X H Y H are homotopic for all i,j and all subgroups H ⊆ G.

How to cite

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Arkowitz, Martin, and Golasiński, Marek. "Co-H-structures on equivariant Moore spaces." Fundamenta Mathematicae 146.1 (1994): 59-67. <http://eudml.org/doc/212051>.

@article{Arkowitz1994,
abstract = {Let G be a finite group, $\mathbb \{O\}_G$ the category of canonical orbits of G and $A : \mathbb \{O\}_G → \mathbb \{A\}$b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with $Ext^\{n-1\}(A, A ⊗ A)$. Then the case $G = ℤ_\{p^k\}$ leads to an example of infinitely many G-homotopically distinct G-maps $φ_i : X → Y$ such that $φ_i^H$, $φ_j^H : X^H → Y^H$ are homotopic for all i,j and all subgroups H ⊆ G.},
author = {Arkowitz, Martin, Golasiński, Marek},
journal = {Fundamenta Mathematicae},
keywords = {homotopy classes of comultiplications; Moore -space},
language = {eng},
number = {1},
pages = {59-67},
title = {Co-H-structures on equivariant Moore spaces},
url = {http://eudml.org/doc/212051},
volume = {146},
year = {1994},
}

TY - JOUR
AU - Arkowitz, Martin
AU - Golasiński, Marek
TI - Co-H-structures on equivariant Moore spaces
JO - Fundamenta Mathematicae
PY - 1994
VL - 146
IS - 1
SP - 59
EP - 67
AB - Let G be a finite group, $\mathbb {O}_G$ the category of canonical orbits of G and $A : \mathbb {O}_G → \mathbb {A}$b a contravariant functor to the category of abelian groups. We investigate the set of G-homotopy classes of comultiplications of a Moore G-space of type (A,n) where n ≥ 2 and prove that if such a Moore G-space X is a cogroup, then it has a unique comultiplication if dim X < 2n - 1. If dim X = 2n-1, then the set of comultiplications of X is in one-one correspondence with $Ext^{n-1}(A, A ⊗ A)$. Then the case $G = ℤ_{p^k}$ leads to an example of infinitely many G-homotopically distinct G-maps $φ_i : X → Y$ such that $φ_i^H$, $φ_j^H : X^H → Y^H$ are homotopic for all i,j and all subgroups H ⊆ G.
LA - eng
KW - homotopy classes of comultiplications; Moore -space
UR - http://eudml.org/doc/212051
ER -

References

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  1. [A-G] M. Arkowitz and M. Golasiński, Co-H-structures on Moore spaces of type (A, 2), Canad. J. Math., to appear. Zbl0829.55006
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  12. [Ka2] P. J. Kahn, Equivariant homology decompositions, ibid., 273-287. 
  13. [Ka3] P. J. Kahn, Steenrod's problem and k-invariants of certain classifying spaces, in: Algebraic K-Theory, Lecture Notes in Math. 967, Springer, 1982, 195-214. 
  14. [Ma] T. Matumoto, On G-CW complexes and a theorem of J. H. C. Whitehead, J. Fac. Sci. Univ. Tokyo 18 (1971), 363-374. 
  15. [M-T] R. E. Mosher and M. C. Tangora, Cohomology Operations and Applications in Homotopy Theory, Harper & Row, New York, 1968. Zbl0153.53302
  16. [Qu] D. G. Quillen, Homotopical Algebra, Lecture Notes in Math. 43, Springer, 1967. 
  17. [Sm] J. R. Smith, Equivariant Moore spaces II - The low dimensional case, J. Pure Appl. Algebra 36 (1985), 187-204. Zbl0561.55017
  18. [Tr] G. V. Triantafillou, Rationalization of Hopf G-spaces, Math. Z. 182 (1983), 485-500. Zbl0518.55008
  19. [Un] H. Unsöld, Topological minimal algebras and Sullivan-de Rham equivalence, Astérisque 113-114 (1984), 337-343. 

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