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stable orthogonal group
Contents
Contents
Definition
For each there is an inclusion
of the orthogonal group in dimension into that in dimension . The stable orthogonal group is the direct limit over this sequence of inclusions.
Properties
Homotopy groups
By the discussion at orthogonal group – homotopy groups we have that the homotopy groups of the stable orthogonal group are
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|
| | | 0 | | 0 | 0 | 0 | |
or if we instead write down the order:
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|
| 2 | 2 | 1 | | 1 | 1 | 1 | |
Via the J-homomorphism this is related to the stable homotopy groups of spheres:
| | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|
Whitehead tower of orthogonal group | | orientation | spin | | string | | | | fivebrane | | | | ninebrane | | | | | |
homotopy groups of stable orthogonal group | | | | 0 | | 0 | 0 | 0 | | | | 0 | | 0 | 0 | 0 | | |
stable homotopy groups of spheres | | | | | | 0 | 0 | | | | | | | 0 | | | | |
image of J-homomorphism | | 0 | | 0 | | 0 | 0 | 0 | | | | 0 | | 0 | 0 | 0 | | |
Last revised on July 14, 2017 at 05:16:51.
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