# 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

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