Algebraic & Geometric Topology

On the metastable homotopy of mod $2$ Moore spaces

Roman Mikhailov 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 study the exponents of metastable homotopy of mod 2 Moore spaces. We prove that the double loop space of 4n–dimensional mod 2 Moore spaces has a multiplicative exponent 4 below the range of 4 times the connectivity. As a consequence, the homotopy groups of 4n–dimensional mod 2 Moore spaces have an exponent of 4 below the range of 4 times the connectivity.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 3 (2016), 1773-1797.

Dates
Received: 2 June 2015
Revised: 20 September 2015
Accepted: 13 October 2015
First available in Project Euclid: 28 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1511895861

Digital Object Identifier
doi:10.2140/agt.2016.16.1773

Mathematical Reviews number (MathSciNet)
MR3523054

Zentralblatt MATH identifier
1354.55005

Subjects
Primary: 55Q52: Homotopy groups of special spaces 14F35: Homotopy theory; fundamental groups [See also 14H30] 55Q20: Homotopy groups of wedges, joins, and simple spaces
Secondary: 55Q05: Homotopy groups, general; sets of homotopy classes 55P35: Loop spaces

Keywords
Moore space Cohen group homotopy groups homotopy exponent metastable homotopy loop space

Citation

Mikhailov, Roman; Wu, Jie. On the metastable homotopy of mod $2$ Moore spaces. Algebr. Geom. Topol. 16 (2016), no. 3, 1773--1797. doi:10.2140/agt.2016.16.1773. https://projecteuclid.org/euclid.agt/1511895861


Export citation

References

  • M,G Barratt, Homotopy ringoids and homotopy groups, Quart. J. Math., Oxford Ser. 5 (1954) 271–290
  • M,G Barratt, Spaces of finite characteristic, Quart. J. Math. Oxford Ser. 11 (1960) 124–136
  • M,G Barratt, M,E Mahowald, The metastable homotopy of ${\rm O}(n)$, Bull. Amer. Math. Soc. 70 (1964) 758–760
  • H,J Baues, Quadratic functors and metastable homotopy, J. Pure Appl. Algebra 91 (1994) 49–107
  • C-F B ödigheimer, Stable splittings of mapping spaces, from: “Algebraic topology”, (H,R Miller, D,C Ravenel, editors), Lecture Notes in Math. 1286, Springer, Berlin (1987) 174–187
  • F,R Cohen, The unstable decomposition of $\Omega \sp{2}\Sigma \sp{2}X$ and its applications, Math. Z. 182 (1983) 553–568
  • F,R Cohen, A course in some aspects of classical homotopy theory, from: “Algebraic topology”, (H,R Miller, D,C Ravenel, editors), Lecture Notes in Math. 1286, Springer, Berlin (1987) 1–92
  • F,R Cohen, On combinatorial group theory in homotopy, from: “Homotopy theory and its applications”, (A Adem, R,J Milgram, D,C Ravenel, editors), Contemp. Math. 188, Amer. Math. Soc. (1995) 57–63
  • F,R Cohen, R Mikhailov, J Wu, A combinatorial approach to the exponents of Moore spaces, preprint (2015)
  • F,R Cohen, J Wu, A remark on the homotopy groups of $\Sigma\sp n{\bf R}{\rm P}\sp 2$, from: “The Čech centennial”, (M Cenkl, H Miller, editors), Contemp. Math. 181, Amer. Math. Soc. (1995) 65–81
  • E Dyer, R,K Lashof, Homology of iterated loop spaces, Amer. J. Math. 84 (1962) 35–88
  • P,S Green, R,A Holzsager, Secondary operations in $K$–theory and applications to metastable homotopy, Illinois J. Math. 16 (1972) 415–422
  • I,M James, On the iterated suspension, Quart. J. Math., Oxford Ser. 5 (1954) 1–10
  • M Mahowald, Some Whitehead products in $S\sp{n}$, Topology 4 (1965) 17–26
  • M Mahowald, The metastable homotopy of $S\sp{n}$, Mem. Amer. Math. Soc. 72, Providence, R.I. (1967)
  • M Mahowald, On the metastable homotopy of ${\rm O}(n)$, Proc. Amer. Math. Soc. 19 (1968) 639–641
  • K Morisugi, Metastable homotopy groups of ${\rm Sp}(n)$, J. Math. Kyoto Univ. 27 (1987) 367–380
  • K Morisugi, J Mukai, Lifting to mod 2 Moore spaces, J. Math. Soc. Japan 52 (2000) 515–533
  • D,A Tipple, A note on the metastable homotopy groups of torsion spheres, Bull. London Math. Soc. 3 (1971) 303–306
  • J Wu, On combinatorial calculations for the James–Hopf maps, Topology 37 (1998) 1011–1023
  • J Wu, Homotopy theory of the suspensions of the projective plane, Mem. Amer. Math. Soc. 769, Providence, RI (2003)
  • J Wu, On maps from loop suspensions to loop spaces and the shuffle relations on the Cohen groups, Mem. Amer. Math. Soc. 851, Providence, RI (2006)