Presentations of the first homotopy groups of the unitary groups

@article{Pttmann2003PresentationsOT,
  title={Presentations of the first homotopy groups of the unitary groups
},
  author={Thomas P{\"u}ttmann and Alcib{\'i}ades Rigas},
  journal={Commentarii Mathematici Helvetici},
  year={2003},
  volume={78},
  pages={648-662}
}
AbstractWe describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n ≥ 2 we supply n homotopic maps that each represent the (n - 1)!-th power of a suitable generator of π2n SU(n) ≈ ℤn!. The product of these n commuting maps is the constant map to the identity matrix.  

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References

SHOWING 1-10 OF 31 REFERENCES
Complex Reections and Polynomial Generators of Homotopy Groups
Starting from suitable maps of the form :S n !SU(m) , for large m , we obtain functions :S n !SU(k) that generate n SU(k) , for certain n and k . We have used this method in elementary algebraicExpand
Explicit construction of nontrivial elements for homotopy groups of classical Lie groups
Nontrivial elements of homotopy groups for unitary, orthogonal, and symplectic groups are given explicitly. In particular, (a) representatives of generators of nontrivial homotopy groups of stableExpand
THE STABLE HOMOTOPY OF THE CLASSICAL GROUPS.
  • R. Bott
  • Mathematics, Medicine
  • Proceedings of the National Academy of Sciences of the United States of America
  • 1957
TLDR
The index of s, denoted by X(s), is the properly counted sum of the conjugate points of P in the interior of s which occurs as the index of some geodesic from P to Q in the class h. Expand
On the parallelizability of the spheres
is always divisible by (2k — 1)!. I wonder if you have noted the connection of this result with classical problems, such as the existence of division algebras, and the parallelizability of spheres.Expand
Topology of Fibre Bundles
Fibre bundles, an integral part of differential geometry, are also important to physics. This text, a succint introduction to fibre bundles, includes such topics as differentiable manifolds andExpand
TOPOLOGY, ALGEBRA, ANALYSIS: RELATIONS AND MISSING LINKS
T his is largely, but not entirely, a historical survey. It puts various matters together that are usually considered in separate contexts. Moreover, it leads to open, probably quite difficultExpand
...
1
2
3
4
...