## Journal of the Mathematical Society of Japan

### Homotopy groups of the spaces of self-maps of Lie groups

#### Abstract

We compute the homotopy groups of the spaces of self maps of Lie groups of rank 2, $\SU(3)$, $\Sp(2)$, and $G_2$. We use the cell structures of these Lie groups and the standard methods of homotopy theory.

#### Article information

Source
J. Math. Soc. Japan, Volume 60, Number 3 (2008), 767-792.

Dates
First available in Project Euclid: 4 August 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1217884492

Digital Object Identifier
doi:10.2969/jmsj/06030767

Mathematical Reviews number (MathSciNet)
MR2440413

Zentralblatt MATH identifier
1148.55002

Subjects
Primary: 55Q05: Homotopy groups, general; sets of homotopy classes
Secondary: 55P10: Homotopy equivalences

#### Citation

MARUYAMA, Ken-ichi; ŌSHIMA, Hideaki. Homotopy groups of the spaces of self-maps of Lie groups. J. Math. Soc. Japan 60 (2008), no. 3, 767--792. doi:10.2969/jmsj/06030767. https://projecteuclid.org/euclid.jmsj/1217884492

#### References

• M. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Proc. Soc. London, A308 (1982), 523–615.
• M. Arkowitz, H. Ōshima and J. Strom, Noncommutativity of the group of self-homotopy classes of classical simple Lie groups, Topology Appl., 125 (2002), 87–96.
• W. Browder and E. Spanier, $H$-spaces and duality, Pacific J. Math., 12 (1970), 411–414.
• G. Didierjean, Homotopie de l'espace des equivalences d'homotopie, Trans. Amer. Math. Soc., 330 (1992), 153–163.
• H. Hamanaka and A. Kono, Homotopy type of gauge groups of $\SU(3)$-bundles over $\s^6$, Topology and its Appl., 154 (2007), 1377–1380.
• H. Kachi, J. Mukai, T. Nozaki, Y. Sumita and D. Tamaki, Some cohomotopy groups of suspended projective planes, Math. J. Okayama Univ., 43 (2001), 105–121.
• A. Kono and H. Ōshima, Commutativity of the group of self homotopy classes of Lie groups, Bull. London Math. Soc., 36 (2004), 37–52.
• M. Mimura, The homotopy groups of Lie groups of low rank, J. Math. Kyoto Univ., 6 (1967), 131–176.
• M. Mimura and H. Ōshima, Self homotopy groups of Hopf spaces with at most three cells, J. Math. Soc. Japan, 51 (1999), 71–92.
• M. Mimura and N. Sawashita, On the group of self-homotopy equivalences of $H$-spaces of rank 2, J. Math. Kyoto Univ., 21 (1981), 331–349.
• M. Mimura and H. Toda, Homotopy groups of $\SU(3)$, $\SU(4)$ and $\Sp(2)$, J. Math. Kyoto Univ., 3 (1964), 217–250.
• J. Mukai, Stable homotopy of some elementary complexes, Mem. Fac. Sci. Kyushu Univ., XX (1966), 266–282.
• H. Ōshima, Self homotopy set of a Hopf space, Quart. J. Math., 50 (1999), 483–495.
• H. Ōshima, Self homotopy group of the exceptional Lie group $G_2$, J. Math. Kyoto Univ., 40 (2000), 177–184.
• H. Ōshima, The group of self-homotopy classes of $SO(4)$, J. Pure App. Algebra, 185 (2003), 193–205.
• H. Toda, A topological proof of theorems of Bott and Borel-Hirzebruch for homotopy groups of unitary groups, Mem. Coll. Sci. Univ. Kyoto, 32 (1959), 103–119.
• H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Studies, 49, Princeton, 1962.