Algebraic & Geometric Topology

On mod $p$ $A_p$–spaces

Ruizhi Huang and Jie Wu

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove a necessary condition for the existence of an Ap–structure on modp spaces, and also derive a simple proof for the finiteness of the number of modp Ap–spaces of given rank. As a direct application, we compute a list of possible types of rank 3 modp homotopy associative H–spaces.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2125-2144.

Dates
Received: 11 February 2016
Revised: 9 December 2016
Accepted: 26 December 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841439

Digital Object Identifier
doi:10.2140/agt.2017.17.2125

Mathematical Reviews number (MathSciNet)
MR3685604

Zentralblatt MATH identifier
1373.55013

Subjects
Primary: 55P45: $H$-spaces and duals 55S25: $K$-theory operations and generalized cohomology operations [See also 19D55, 19Lxx]
Secondary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 55P15: Classification of homotopy type 55S05: Primary cohomology operations

Keywords
$A_p$-space $\psi$-operation homotopy associative $H$-space Steenrod powers Adem relations

Citation

Huang, Ruizhi; Wu, Jie. On mod $p$ $A_p$–spaces. Algebr. Geom. Topol. 17 (2017), no. 4, 2125--2144. doi:10.2140/agt.2017.17.2125. https://projecteuclid.org/euclid.agt/1510841439


Export citation

References

  • J,F Adams, On the non-existence of elements of Hopf invariant one, Ann. of Math. 72 (1960) 20–104
  • J,F Adams, The sphere, considered as an $H$–space ${\rm mod}\,p$, Quart. J. Math. Oxford. Ser. 12 (1961) 52–60
  • J,F Adams, $H$–spaces with few cells, Topology 1 (1962) 67–72
  • J,F Adams, M,F Atiyah, $K$–theory and the Hopf invariant, Quart. J. Math. Oxford Ser. 17 (1966) 31–38
  • M,F Atiyah, Power operations in $K$–theory, Quart. J. Math. Oxford Ser. 17 (1966) 165–193
  • R,A Body, R,R Douglas, Homotopy types within a rational homotopy type, Topology 13 (1974) 209–214
  • R,R Douglas, F Sigrist, Sphere bundles over spheres and $H$–spaces, Topology 8 (1969) 115–118
  • J Grbić, J Harper, M Mimura, S Theriault, J Wu, Rank $p-1$ mod-$p$ $H$–spaces, Israel J. Math. 194 (2013) 641–688
  • N,L Hagelgans, Local spaces with three cells as $H$–spaces, Canad. J. Math. 31 (1979) 1293–1306
  • J,R Harper, The ${\rm mod}\ 3$ homotopy type of $F\sb{4}$, from “Localization in group theory and homotopy theory, and related topics” (P Hilton, editor), Lecture Notes in Math. 418, Springer (1974) 58–67
  • J Harper, A Zabrodsky, Evaluating a $p\sp{\mathit{th}}$ order cohomology operation, Publ. Mat. 32 (1988) 61–78
  • Y Hemmi, Homotopy associative finite $H$–spaces and the ${\rm mod}\ 3$ reduced power operations, Publ. Res. Inst. Math. Sci. 23 (1987) 1071–1084
  • Y Hemmi, On exterior $A\sb{n}$–spaces and modified projective spaces, Hiroshima Math. J. 24 (1994) 583–605
  • Y Hemmi, Mod $p$ decompositions of mod $p$ finite $H$–spaces, Mem. Fac. Sci. Kochi Univ. Ser. A Math. 22 (2001) 59–65
  • J,R Hubbuck, Generalized cohomology operations and $H$–spaces of low rank, Trans. Amer. Math. Soc. 141 (1969) 335–360
  • J,R Hubbuck, M Mimura, The number of mod $p\ A(p)$–spaces, Illinois J. Math. 33 (1989) 162–169
  • N Iwase, On the $K$–ring structure of $X$–projective $n$–space, Mem. Fac. Sci. Kyushu Univ. Ser. A 38 (1984) 285–297
  • J McCleary, Mod $p$ decompositions of $H$–spaces; another approach, Pacific J. Math. 87 (1980) 373–388
  • M Mimura, G Nishida, H Toda, On the classification of $H$–spaces of rank $2$, J. Math. Kyoto Univ. 13 (1973) 611–627
  • M Mimura, G Nishida, H Toda, ${\rm Mod}\ p$ decomposition of compact Lie groups, Publ. Res. Inst. Math. Sci. 13 (1977/78) 627–680
  • J,D Stasheff, Homotopy associativity of $H$–spaces, I, Trans. Amer. Math. Soc. 108 (1963) 275–292
  • J,D Stasheff, Homotopy associativity of $H$–spaces, II, Trans. Amer. Math. Soc. 108 (1963) 293–312
  • J,D Stasheff, The ${\rm mod}\ p$ decomposition of Lie groups, from “Localization in group theory and homotopy theory, and related topics” (P Hilton, editor), Lecture Notes in Math. 418, Springer (1974) 142–149
  • C Wilkerson, $K$–theory operations in ${\rm mod}$ $p$ loop spaces, Math. Z. 132 (1973) 29–44
  • C Wilkerson, Spheres which are loop spaces ${\rm mod}$ $p$, Proc. Amer. Math. Soc. 39 (1973) 616–618
  • C Wilkerson, ${\rm Mod}$ $p$ decomposition of ${\rm mod}$ $p$ $H$–spaces, from “Algebraic and geometrical methods in topology” (L,F McAuley, editor), Lecture Notes in Math. 428, Springer (1974) 52–57
  • A Zabrodsky, Homotopy associativity and finite ${\rm CW}$ complexes, Topology 9 (1970) 121–128
  • A Zabrodsky, The classification of simply connected $H$–spaces with three cells, I, Math. Scand. 30 (1972) 193–210
  • A Zabrodsky, The classification of simply connected $H$–spaces with three cells, II, Math. Scand. 30 (1972) 211–222
  • A Zabrodsky, On the realization of invariant subgroups of $\pi \sb\ast (X)$, Trans. Amer. Math. Soc. 285 (1984) 467–496