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

top
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 = 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.

How to cite

top

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

top
  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
  2. [Br] G. E. Bredon, Equivariant Cohomology Theories, Lecture Notes in Math. 34, Springer, 1967. 
  3. [Ca] G. Carlsson, A counterexample to a conjecture of Steenrod, Invent. Math. 64 (1981), 171-174. Zbl0477.55007
  4. [Co] S. R. Costenoble and S. Waner, A nonexistence result for Moore G-spectra, Proc. Amer. Math. Soc. 113 (1991), 265-274. 
  5. [Do1] R. Doman, Non-G-equivalent Moore G-spaces of the same type, ibid. 103 (1988), 1317-1321. Zbl0666.55007
  6. [Do2] R. Doman, Moore G-spaces which are not co-Hopf G-spaces, Canad. Math. Bull. 32 (1989), 365-368. Zbl0645.55007
  7. [D-D-K] E. Dror, Dwyer and D. M. Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), 670-672. Zbl0454.55018
  8. [El] A. D. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983), 275-284. 
  9. [Il1] S. Illman, Equivariant algebraic topology, Ph. D. Thesis, Princeton University, Princeton, N.J., 1972. 
  10. [Il2] S. Illman, Equivariant singular homology and cohomology I, Mem. Amer. Math. Soc. 156 (1975). 
  11. [Ka1] P. J. Kahn, Rational Moore G-spaces, Trans. Amer. Math. Soc. 298 (1986), 245-271. Zbl0616.55009
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.