Six-term exact sequence

Remarks

Half-exactness of $K_0$ and $K_1$ together with the index map give an exact sequence $K_1(I) \to K_1(A) \to K_1(B) \to K_0(I) \to K_0(A) \to K_0(B)$, which continues to the left with the groups $K_n = K_0 \circ S^n$ of suspensions. Bott periodicity identifies $K_{n+2}$ with $K_n$, and the long sequence folds into six terms; the exponential map is the index map of the suspended sequence composed with the Bott map. Example: for $0 \to \mathcal{K} \to \mathcal{T} \to C(\mathbb{T}) \to 0$, the index map sends the generator $[z]$ of $K_1(C(\mathbb{T})) \cong \Z$ to a generator of $K_0(\mathcal{K}) \cong \Z$, and the sequence then gives $K_0(\mathcal{T}) \cong \Z$ and $K_1(\mathcal{T}) = 0$. The sequence needs a genuine short exact sequence: exactness in the middle, $\ker \pi = \iota(I)$, is used to build the connecting maps.

Going deeper

For a student

This is the main computational tool of operator K-theory, the analogue of the long exact sequence in homology for a pair of spaces. For $A = C(X)$ and $I = C_0(X \setminus Y)$ with $Y$ a closed subset, it is the six-term sequence of topological K-theory for the pair $(X, Y)$. The index map generalizes the Fredholm index. The exponential map is named after its formula. For unital $A$, let $p$ be a projection in $M_n(B)$ with self-adjoint lift $h \in M_n(A)$. Then $e^{2 \pi i h}$ is a unitary over the unitization of $I$, and its class gives $\partial_0[p]$ up to a sign convention. The Pimsner–Voiculescu sequence for crossed products by $\Z$ is derived from it.