Bott periodicity theorem

A fundamental theorem in $ K $- theory which, in its simplest form, states that for any (compact) space $ X $ there exists an isomorphism between the rings $ K(X) \otimes K(S ^ {2} ) $ and $ K(X \times S ^ {2} ) $. More generally, if $ L $ is a complex vector bundle over $ X $ and $ P(L \oplus 1) $ is the projectivization of $ L \oplus 1 $, then the ring $ K(P(L \oplus 1)) $ is a $ K(X) $- algebra with one generator $ [H] $ and a unique relation $ ([H] - [1])([L][H] - [1]) = 0 $, where $ [E] $ is the image of a vector bundle $ E $ in $ K(X) $ and $ H ^ {-1} $ is the Hopf fibration over $ P(L \oplus 1) $. This fact is equivalent to the existence of a Thom isomorphism in $ K $- theory for complex vector bundles. In particular, $ P(1 \oplus 1) = X \times S ^ {2} $. Bott's periodicity theorem was first demonstrated by R. Bott [1] using Morse theory, and was then re-formulated in terms of $ K $- theory [6]; an analogous theorem has also been demonstrated for real fibre bundles.

Bott's periodicity theorem establishes the property of the stable homotopy type of the unitary group $ U $, consisting in the fact that $ {\Omega ^ {2} } U \sim U $, where $ \Omega X $ is the space of loops on $ X $, and $ \sim $ is weak homotopy equivalence, in particular $ \pi _ {i} (U) = \pi _ {i+2} (U) $ for $ i = 0, 1 \dots $ where $ \pi _ {i} $ is the $ i $- th homotopy group. Similarly, for the orthogonal group $ O $:

$$ \Omega ^ {8} O \sim O,\ \ \pi _ {i} (O) = \pi _ {i+ 8 } (O). $$

References

[1]	R. Bott, "The stable homotopy of the classical groups" Ann. of Math. (2) , 70 : 2 (1959) pp. 313–337 MR0110104 Zbl 0129.15601
[2]	J.W. Milnor, "Morse theory" , Princeton Univ. Press (1963) MR0163331 Zbl 0108.10401
[3]	M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967) MR224083
[4]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966) MR0229247 Zbl 0144.44804
[5]	J.C. Moore, "On the periodicity theorem for complex vector bundles" , Sem. H. Cartan (1959–1960)
[6]	M.F. Atiyah, R. Bott, "On the periodicity theorem for complex vector bundles" Acta Math. , 112 (1964) pp. 229–247 MR0178470 Zbl 0131.38201

References

[a1]	R. Bott, "Lectures on $K(X)$" , Benjamin (1969) MR0258020 Zbl 0194.23904
[a2]	M. Karoubi, "$K$-theory" , Springer (1978) MR0488029 Zbl 0382.55002