The Bott periodicity theorem states that the homotopy groups of the stable orthogonal group and symplectic group repeat with period 8, while those of the stable unitary group repeat with period 2 (Bott 1959).
More precisely, let
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(1)
|
 |  |  |
(2)
|
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(3)
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where the direct limits are taken with respect to the standard block inclusions. A space-level form of the theorem is
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(4)
|
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(5)
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(6)
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where 
is the based loop space of 
, and 
denotes a homotopy equivalence.
Equivalently, for 
,
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(7)
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(8)
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(9)
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The stable homotopy groups of 
are therefore
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(10)
|
while those of 
are
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(11)
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and those of 
are shifted by four dimensions,
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(12)
|
The theorem also underlies the periodic classification of stable complex and real vector bundles (Atiyah 1967).
