Consider six countable Abelian groups and six group homomorphims as in the following diagram
G → H → I ↑ ↓ L ← K ← J
Assume that the resulting sequence is exact at all six entries.
Question: Is there a (second countable) locally compact Hausdorff space X with a closed subspace A, such that the resulting six-term sequence in K-theory
K0(X,A) → K0(X) → K0(A) ↑ ↓ K1(A) ← K1(X) ← K1(X,A)
is isomorphic to the above one?
An answer in the finitely generated case would also be interesting.