Context and Significance
The context of Bott periodicity is that the homotopy groups of spheres, which would be expected to play the basic part in algebraic topology by analogy with homology theory, have proved elusive (and the theory is complicated). The subject of stable homotopy theory was conceived as a simplification, by introducing the suspension (smash product with a circle) operation, and seeing what (roughly speaking) remained of homotopy theory once one was allowed to suspend both sides of an equation, as many times as one wished. The stable theory was still hard to compute with, in practice.
What Bott periodicity offered was an insight into some highly non-trivial spaces, with central status in topology because of the connection of their cohomology with characteristic classes, for which all the (unstable) homotopy groups could be calculated. These spaces are the (infinite, or stable) unitary, orthogonal and symplectic groups U, O and Sp. In this context, stable refers to taking the union U (also known as the direct limit) of the sequence of inclusions
and similarly for O and Sp. Bott's (now somewhat awkward) use of the word stable in the title of his seminal paper refers to these stable classical groups and not to stable homotopy groups.
The important connection of Bott periodicity with the stable homotopy groups of spheres comes via the so called stable J-homomorphism from the (unstable) homotopy groups of the (stable) classical groups to these stable homotopy groups . Originally described by George W. Whitehead, it became the subject of the famous Adams conjecture (1963) which was finally resolved in the affirmative by Daniel Quillen (1971).
Bott's original results may be succinctly summarized in:
Corollary: The (unstable) homotopy groups of the (infinite) classical groups are periodic:
Note: The second and third of these isomorphisms intertwine to give the 8-fold periodicity results:
Read more about this topic: Bott Periodicity Theorem
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