The Art of Toda Brackets-A Journey through the Homotopy Groups of SO(n) in degrees 23-29
Abstract
This book has several fundamental aims, one of which is to serve as a reference book for computing unstable homotopy groups using methods of the Toda school. The Toda bracket serves the art of constructing homotopy liftings and extensions of maps. It plays a fundamental role in dealing with composition relations of homotopy classes. This book brings together in Appendix A.3 all known results on the $2$-primary components of unstable homotopy groups of spheres obtained using the art of Toda brackets as the principal tool, with explicit generators. It also explains in Appendix A.2 the conventions used to name the generators. The book mainly presents calculations of the $2$-primary components of the homotopy groups πi(SO(n)) for all 23≤ i≤ 28 and all n, as well as some π29(SO(n)), with explicit generators, while retaining two possible group structures for π28(SO(11)). In degree $29,$ the cases 7≤ n≤14 remain under investigation. The odd-primary components of the homotopy groups πi(SO(n)) in the above range can be computed with little effort; these computations essentially amount to looking up the relevant entries in existing tables.
Related
- Homotopy groups of spheres and dimension quotients
- Day Convolution and Homotopy Groups on the Pointed Homotopy Category of CW Complexes
- The zeroth stable homotopy groups of motivic spheres over the integers
- Medvedev degrees of SFTs on cocompact Fuchsian groups
- Vanishing-set inclusion and character degrees in finite p-groups
BibTeX
@misc{hirato2026toda,
title = {{The Art of Toda Brackets-A Journey through the Homotopy Groups of SO(n) in degrees 23-29}},
author = {Yoshihiro Hirato and Jin-ho Lee and Toshiyuki Miyauchi and Juno Mukai and Mariko Ohara and Juxin Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28447},
eprint = {2610.28447},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28447},
}
RIS
TY - UNPB
TI - The Art of Toda Brackets-A Journey through the Homotopy Groups of SO(n) in degrees 23-29
AU - Yoshihiro Hirato
AU - Jin-ho Lee
AU - Toshiyuki Miyauchi
AU - Juno Mukai
AU - Mariko Ohara
AU - Juxin Yang
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.28447
UR - https://arxiv.org/abs/2610.28447
AB - This book has several fundamental aims, one of which is to serve as a reference book for computing unstable homotopy groups using methods of the Toda school. The Toda bracket serves the art of constructing homotopy liftings and extensions of maps. It plays a fundamental role in dealing with composition relations of homotopy classes. This book brings together in Appendix A.3 all known results on the $2$-primary components of unstable homotopy groups of spheres obtained using the art of Toda brackets as the principal tool, with explicit generators. It also explains in Appendix A.2 the conventions used to name the generators. The book mainly presents calculations of the $2$-primary components of the homotopy groups π_i(SO(n)) for all 23≤ i≤ 28 and all n, as well as some π_29(SO(n)), with explicit generators, while retaining two possible group structures for π_28(SO(11)). In degree $29,$ the cases 7≤ n≤14 remain under investigation. The odd-primary components of the homotopy groups π_i(SO(n)) in the above range can be computed with little effort; these computations essentially amount to looking up the relevant entries in existing tables.
ER -