Exponents of \([\Omega ({\mathbb {S}}^{r+1}), \Omega (Y)]\)

  • Marek Golasiński
  • Daciberg Lima Gonçalves
  • Peter WongEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)


We investigate the exponents of the total Cohen groups \([\Omega ({\mathbb S}^{r+1}), \Omega (Y)]\) for any \(r\ge 1\). In particular, we show that for \(p\ge 3\), the p-primary exponents of \([\Omega ({\mathbb S}^{r+1}), \Omega ({\mathbb S}^{2n+1})]\) and \([\Omega ({\mathbb S}^{r+1}), \Omega ({\mathbb S}^{2n})]\) coincide with the p-primary homotopy exponents of spheres \({\mathbb S}^{2n+1}\) and \({\mathbb S}^{2n}\), respectively. We further study the exponent problem when Y is a space with the homotopy type of \(\Sigma (n)/G\) for a homotopy n-sphere \(\Sigma (n)\), the complex projective space \(\mathbb {C}P^n\) for \(n\ge 1\) or the quaternionic projective space \(\mathbb {H}P^n\) for \(1\le n\le \infty \).


Barratt-Puppe sequence Cohen group EHP sequence James construction James-Hopf map (invariant) Moore space p-primary (homotopy) exponent Projective space Homotopy space form Whitehead product 

2010 Mathematics Subject Classication

Primary: 55Q05 55Q15 55Q20 Secondary: 55P65 



This work was initiated and completed during the authors’ visits to Banach Center in Warsaw, Poland, October 27–November 05, 2016 and February 17–March 03, 2018, respectively. The authors would like to thank the Banach Center in Warsaw, Poland, and the Faculty of Mathematics and Computer Science, the University of Warmia and Mazury in Olsztyn, Poland, for their hospitality and support.

Special thanks are due to Jim Stasheff for pointing out the Dold–Lashof result in [4, 23] and to Jie Wu for helpful conversations regarding the Cohen groups. Finally, the authors would like to express their gratitude to three anonymous referees for their invaluable suggestions which help improve the exposition of the paper.


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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Marek Golasiński
    • 1
  • Daciberg Lima Gonçalves
    • 2
  • Peter Wong
    • 3
    Email author
  1. 1.Faculty of Mathematics and Computer ScienceUniversity of Warmia and MazuryOlsztynPoland
  2. 2.Department of Mathematics - IME - USP, Rua do Matão 1010 CEPSão PauloBrazil
  3. 3.Department of Mathematics, Bates CollegeLewistonUSA

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