Thanks for drawing my attention to the long exact sequence*; if I'm interpreting it correctly, the injection at 𝛼 and projection at 𝛽 split, so we have a direct sum: H(X,∅) ≃ H(X,A) ⊕ H(A,∅)?

(never mind the function analogy, I was trying to handwave a quotient in the other direction but that would fall immediately out of the direct sum if I understand correctly: A ≃ (A⊕B)/B and B ≃ (A⊕B)/A?)

* which forms a Möbius band in its own way, because at H(A,∅) we feed it into 𝛼 as ∅ ⊕ Cycles(A) but get it out of 𝛾 as Cycles(A) ⊕ ∅, leading to a "twist"?

[Edit: are you aware of Dan Piponi's blogging?

http://blog.sigfpe.com/2006/08/algebraic-topology-in-haskell...

http://blog.sigfpe.com/2006/08/what-can-we-measure-part-i.ht...

http://blog.sigfpe.com/2010/01/target-enumeration-with-euler...

etc.]