Topological realization of a family of pseudoreflection groups

Dietrich Notbohm

Fundamenta Mathematicae (1998)

  • Volume: 155, Issue: 1, page 1-31
  • ISSN: 0016-2736

Abstract

top
We are interested in a topological realization of a family of pseudoreflection groups G G L ( n , F p ) ; i.e. we are looking for topological spaces whose mod-p cohomology is isomorphic to the ring of invariants F p [ x 1 , . . . , x n ] G . Spaces of this type give partial answers to a problem of Steenrod, namely which polynomial algebras over F p can appear as the mod-p cohomology of a space. The family under consideration is given by pseudoreflection groups which are subgroups of the wreath product / q Σ n where q divides p - 1 and where p is odd. Let G be such a subgroup acting on the polynomial algebra A : = F p [ x 1 , . . . , x n ] . We show that there exists a space X such that H * ( X ; F p ) A G which is again a polynomial algebra. Examples of polynomial algebras of this form are given by the mod-p cohomology of the classifying spaces of special orthogonal groups or of symplectic groups.  The construction uses products of classifying spaces of unitary groups as building blocks which are glued together via information encoded in a full subcategory of the orbit category of the group G. Using this construction we also show that the homotopy type of the p-adic completion of these spaces is completely determined by the mod-p cohomology considered as an algebra over the Steenrod algebra. Moreover, we calculate the set of homotopy classes of self maps of the completed spaces.

How to cite

top

Notbohm, Dietrich. "Topological realization of a family of pseudoreflection groups." Fundamenta Mathematicae 155.1 (1998): 1-31. <http://eudml.org/doc/212240>.

@article{Notbohm1998,
abstract = {We are interested in a topological realization of a family of pseudoreflection groups $G ⊂ GL(n,\{F\}_p )$; i.e. we are looking for topological spaces whose mod-p cohomology is isomorphic to the ring of invariants $\{F\}_p [x_1,..., x_n]^G$. Spaces of this type give partial answers to a problem of Steenrod, namely which polynomial algebras over $\{F\}_p $ can appear as the mod-p cohomology of a space. The family under consideration is given by pseudoreflection groups which are subgroups of the wreath product $ℤ/q ≀Σ_n$ where q divides p - 1 and where p is odd. Let G be such a subgroup acting on the polynomial algebra $A:= \{F\}_p [x_1,..., x_n]$. We show that there exists a space X such that $H*(X;\{F\}_p )≅ A^G$ which is again a polynomial algebra. Examples of polynomial algebras of this form are given by the mod-p cohomology of the classifying spaces of special orthogonal groups or of symplectic groups.  The construction uses products of classifying spaces of unitary groups as building blocks which are glued together via information encoded in a full subcategory of the orbit category of the group G. Using this construction we also show that the homotopy type of the p-adic completion of these spaces is completely determined by the mod-p cohomology considered as an algebra over the Steenrod algebra. Moreover, we calculate the set of homotopy classes of self maps of the completed spaces.},
author = {Notbohm, Dietrich},
journal = {Fundamenta Mathematicae},
keywords = {pseudoreflection groups; p-compact group; classifying spaces; compact Lie group; polynomial algebra; -compact group; classifying space; pseudoreflection group},
language = {eng},
number = {1},
pages = {1-31},
title = {Topological realization of a family of pseudoreflection groups},
url = {http://eudml.org/doc/212240},
volume = {155},
year = {1998},
}

TY - JOUR
AU - Notbohm, Dietrich
TI - Topological realization of a family of pseudoreflection groups
JO - Fundamenta Mathematicae
PY - 1998
VL - 155
IS - 1
SP - 1
EP - 31
AB - We are interested in a topological realization of a family of pseudoreflection groups $G ⊂ GL(n,{F}_p )$; i.e. we are looking for topological spaces whose mod-p cohomology is isomorphic to the ring of invariants ${F}_p [x_1,..., x_n]^G$. Spaces of this type give partial answers to a problem of Steenrod, namely which polynomial algebras over ${F}_p $ can appear as the mod-p cohomology of a space. The family under consideration is given by pseudoreflection groups which are subgroups of the wreath product $ℤ/q ≀Σ_n$ where q divides p - 1 and where p is odd. Let G be such a subgroup acting on the polynomial algebra $A:= {F}_p [x_1,..., x_n]$. We show that there exists a space X such that $H*(X;{F}_p )≅ A^G$ which is again a polynomial algebra. Examples of polynomial algebras of this form are given by the mod-p cohomology of the classifying spaces of special orthogonal groups or of symplectic groups.  The construction uses products of classifying spaces of unitary groups as building blocks which are glued together via information encoded in a full subcategory of the orbit category of the group G. Using this construction we also show that the homotopy type of the p-adic completion of these spaces is completely determined by the mod-p cohomology considered as an algebra over the Steenrod algebra. Moreover, we calculate the set of homotopy classes of self maps of the completed spaces.
LA - eng
KW - pseudoreflection groups; p-compact group; classifying spaces; compact Lie group; polynomial algebra; -compact group; classifying space; pseudoreflection group
UR - http://eudml.org/doc/212240
ER -

References

top
  1. [1] J. F. Adams and Z. Mahmud, Maps between classifying spaces, Invent. Math. 35 (1976), 1-41. Zbl0306.55019
  2. [2] J. F. Adams and C. W. Wilkerson, Finite H-spaces and algebras over the Steenrod algebra, Ann. of Math. 111 (1980), 95-143. Zbl0404.55020
  3. [3] J. Aguadé, Constructing modular classifying spaces, Israel J. Math. 66 (1989), 23-40. Zbl0697.55002
  4. [4] J. Aguadé, C. Broto and D. Notbohm, Homotopy classification of some spaces with interesting cohomology and a conjecture of Cooke, Part I, Topology 33 (1994), 455-492. Zbl0843.55007
  5. [5] A. Bousfield and D. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Math. 304, Springer, 1972. Zbl0259.55004
  6. [6] T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups, Springer, 1985. Zbl0581.22009
  7. [7] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press, 1956. 
  8. [8] A. Clark and J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434. Zbl0333.55002
  9. [9] W. Dwyer and D. Kan, Centric maps and realization of diagrams in the homotopy category, Proc. Amer. Math. Soc. 114 (1992), 575-584. Zbl0742.55004
  10. [10] W. Dwyer, H. Miller and C. Wilkerson, Homotopical uniqueness of classifying spaces, Topology 31 (1992), 29-45. Zbl0748.55005
  11. [11] W. G. Dwyer and C. W. Wilkerson, A cohomology decomposition theorem, Topology 31 (1992), 433-443. Zbl0756.55012
  12. [12] W. G. Dwyer and C. W. Wilkerson, A new finite loop space at the prime two, J. Amer. Math. Soc. 6 (1993), 37-63. Zbl0769.55007
  13. [13] W. G. Dwyer and C. W. Wilkerson, Homotopy fixed point methods for Lie groups and finite loop spaces, Ann. of Math. 139 (1994), 395-442. Zbl0801.55007
  14. [14] W. G. Dwyer and C. W. Wilkerson, The center of a p-compact group, in: The Čech Centennial (Boston, Mass., 1993), Contemp. Math. 181, Amer. Math. Soc., 1995, 119-157. Zbl0828.55009
  15. [15] W. Dwyer and A. Zabrodsky, Maps between classifying spaces, in: Algebraic Topology (Barcelona, 1986), Lecture Notes in Math. 1298, Springer, 1987, 106-119. 
  16. [16] P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory, Springer, 1967. Zbl0186.56802
  17. [17] K. Ishiguro, Unstable Adams operations on classifying spaces, Math. Proc. Cambridge Philos. Soc. 102 (1987), 71-75. 
  18. [18] S. Jackowski and J. McClure, Homotopy decomposition of classifying spaces via elementary abelian subgroups, Topology 31 (1992), 113-132. Zbl0754.55014
  19. [19] S. Jackowski, J. McClure and B. Oliver, Homotopy classification of self-maps of BG via G-actions, Ann. of Math. 135 (1992), 183-270. Zbl0771.55003
  20. [20] S. Jackowski, J. McClure and B. Oliver, Self homotopy equivalences of classifying spaces of compact connected Lie groups, Fund. Math. 147 (1995), 99-126. Zbl0835.55012
  21. [21] S. Jackowski, J. McClure and B. Oliver, Homotopy of classifying spaces of compact Lie groups, in: Algebraic Topology and its Applications, Springer, 1994, 81-123. Zbl0796.55009
  22. [22] S. Lang, Algebra, Addison-Wesley, 1965. 
  23. [23] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'un p-groupe abélien élémentaire, Publ. Math. I.H.E.S. 75 (1992), 135-244. 
  24. [24] H. Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. 120 (1984), 39-87. Zbl0552.55014
  25. [25] J. M. Møller, Rational isomorphisms of p-compact groups, Topology 35 (1996), 201-225. Zbl0852.55011
  26. [26] J. M. Møller and D. Notbohm, Centers and finite coverings of finite loop spaces, J. Reine Angew. Math. 456 (1994), 99-113. 
  27. [27] J. M. Møller and D. Notbohm, Connected finite loop spaces with maximal tori, Math. Gott. Heft 14 (1994). 
  28. [28] D. Notbohm, Maps between classifying spaces, Math. Z. 207 (1991), 153-168. Zbl0731.55011
  29. [29] D. Notbohm, Homotopy uniqueness of classifying spaces of compact connected Lie groups at primes dividing the order of the Weyl group, Topology 33 (1994), 271-330. Zbl0820.57020
  30. [30] B. Oliver, Higher limits via Steinberg representations, Comm. Algebra 22 (1994), 1381-1393. Zbl0815.55003
  31. [31] D. Quillen, On the cohomology and K-theory of the general linear groups over a finite field, Ann. of Math. 96 (1972), 552-586. Zbl0249.18022
  32. [32] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274-304. Zbl0055.14305
  33. [33] N. E. Steenrod, Polynomial algebras over the algebra of cohomology operations, in: H-spaces (Neuchâtel, 1970), Lecture Notes in Math. 196, Springer, 1971, 85-99. 
  34. [34] Z. Wojtkowiak, On maps from holim F to Z, in: Algebraic Topology (Barcelona, 1986), Lecture Notes in Math. 1298, Springer, 1987, 227-236. 
  35. [35] C. Xu, The existence and uniqueness of simply connected p-compact groups with Weyl groups W such that |W| is not divisible by the square of p, thesis, Purdue University, 1994. 
  36. [36] A. Zabrodsky, On the realization of invariant subgroups of π * ( X ) , Trans. Amer. Math. Soc. 285 (1984), 467-496. Zbl0576.55009

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.