Abstract
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 \(1\le n\le \infty \).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
H.J. Baues, Commutator Calculus and Groups of Homotopy Classes. London Mathematical Society Lecture Note Series, vol. 50 (Cambridge University Press, Cambridge 1981)
F.R. Cohen, The unstable decomposition of \(\Omega ^2\Sigma ^2X\) and its applications. Math. Z. 182(4), 553–568 (1983)
F.R. Cohen, J. Moore, J. Neisendorfer, The double suspension and exponents of the homotopy groups of spheres. Ann. Math. 110(3), 549–565 (1979)
A. Dold, R. Lashof, Principal quasifibrations and fiber homotopy equivalence of bundles. Ill. J. Math. 3(2), 285–305 (1959)
M. Golasiński, D. Gonçalves, P. Wong, On the group structure of \([\Omega \mathbb{S}^2, \Omega Y]\). Q. J. Math. 66(1), 111–132 (2015)
M. Golasiński, D. Gonçalves, P. Wong, On the group structure of \([J(X), \Omega (Y)]\). J. Homotopy Relat. Struct. 12(3), 707–726 (2017)
M. Golasiński, J. Mukai, Gottlieb and Whitehead Center Groups of Spheres, Projective and Moore Spaces (Springer, Cham, 2014)
B. Gray, On the sphere of origin of infinite families in the homotopy groups of spheres. Topology 8, 219–232 (1969)
J. Grbić, H. Zhao, Homotopy exponents of some homogeneous spaces. Q. J. Math. 62(4), 953–976 (2011)
A. Hatcher, Algebraic Topology (Cambridge University Press, Cambridge, 2002)
P.J. Hilton, On the homotopy groups of the union of spheres. J. Lond. Math. Soc. 30, 154–172 (1955)
I.M. James, The suspension triad of a sphere. Ann. Math. 63(2), 407–429 (1956)
I.M. James, On the suspension sequence. Ann. Math. 65, 74–107 (1957)
J. Long, Thesis. Ph.D. Thesis, Princeton University (1978)
J. Milnor, On spaces having the homotopy type of a \(CW\)-complex. Trans. Am. Math. Soc. 90, 272–280 (1959)
J. Neisendorfer, \(3\)-primary exponents. Math. Proc. Camb. Philos. Soc. 90(1), 63–83 (1981)
J. Neisendorfer, The exponent of a Moore space, in Algebraic Topology and Algebraic K-Theory (Princeton, 1983). Annals of Mathematics Studies, vol. 113, ed. by W. Browder (Princeton University, Princeton, 1987), pp. 35–71
P.S. Selick, Odd primary torsion in \(\pi _k(\mathbb{S}^3)\). Topology 17, 407–412 (1978)
P.S. Selick, \(2\)-primary exponents for the homotopy groups of spheres. Topology 23, 97–99 (1984)
P.J. Selick, J. Wu, On natural coalgebra decompositions of tensor algebras and loop suspensions. Mem. Amer. Math. Soc. 148 (701) (2000) 701
J.-P. Serre, Groups d’homotopie et classes groupes d’homotopie. Ann. Math. 58, 258–294 (1953)
D. Stanley, Exponents and suspensions. Math. Proc. Camb. Philos. Soc. 133(1), 109–116 (2002)
J. Stasheff, H-spacers from a Homotopy Point of View, vol. 161, Lecture Notes in Math (Springer-Verlag, Berlin-Heidelberg-New York, 1970)
S. Theriault, Homotopy exponents of mod \(2^r\) Moore spaces. Topology 47(6), 369–398 (2008)
H. Toda, On the double suspension \(E^2\). Inst. Polytech. Osaka City Univ. Ser. A: 7, 103–145 (1956)
H. Toda, Composition Methods in Homotopy Groups of Spheres. Annals of Mathematics Studies, vol. 49 (Princeton University Press, Princeton, 1962)
G.W. Whitehead, Elements of Homotopy Theory, Graduate Texts in Mathematics 61 (Springer, Berlin, 1978)
J. Wu, Homotopy theory of the suspensions of the projective plane. Mem. Am. Math. Soc. 162, 769 (2003)
J. Wu, On maps from loop suspensions to loop spaces and the shuffle relations on the Cohen groups. Mem. Am. Math. Soc. 180, 851 (2006)
H. Zhao, L. Zhong, W. Shen, Homotopy exponents of Stiefel manifolds in the stable range, Acta Math. Sin. (Engl. Ser.) 32(9), 1080–1088 (2016)
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Golasiński, M., Gonçalves, D.L., Wong, P. (2019). Exponents of \([\Omega ({\mathbb {S}}^{r+1}), \Omega (Y)]\). In: Singh, M., Song, Y., Wu, J. (eds) Algebraic Topology and Related Topics. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-5742-8_7
Download citation
DOI: https://doi.org/10.1007/978-981-13-5742-8_7
Published:
Publisher Name: Birkhäuser, Singapore
Print ISBN: 978-981-13-5741-1
Online ISBN: 978-981-13-5742-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)