I hope this isn't too narrowly focused. I have a question concerning the inverse of the Bott map as defined in Atiyah's paper, *Bott Periodicity and the Index of Elliptic Operators.* On page 122 he defines it as the composition

$$K^{-2}(X) \to K(S^2 \times X) \stackrel{\text{index} \bar\partial}{\longrightarrow} K(X).$$

My question is: what is the first map? At first I thought it was $$ K^{-2}(X) = \tilde K(S^2 \wedge X^+) \to \tilde K(S^2 \times X^+) \to \tilde K(S^2\times X) \subset K(S^2 \times X) $$ where the first map is the pullback of the projection and the second is the pullback of the inclusion. But from the 6 term exact sequence, for this first map to be injective (which it needs to be for this to be correct) we need $$ K(S^2 \times X^+, S^2 \times X) \to \tilde K(S^2 \times X^+) $$ to be the zero map which doesn't seem correct since $S^2 \times X^+/S^2 \times X$ is identifiable with $S^2 \sqcup \{pt\}$ and if we pullback a non-trivial bundle over this space to $S^2 \times X^+$ it seems like it won't trivialize.

Thanks!