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"?

The direct sum decomposition you mention doesn't always happen, as you can see in some of the examples in the calculator. It is closer to happening when \gamma is zero, and if \gamma is zero and we switch from integer coefficient to field coefficients, then it always happens!

obviously I need to play with the calculator more (and probably work the exercises) ... hope you don't mind if I get back to you with more Q's once I have them.

well, I love the subject, so I'm glad to discuss whatever! And let me know if all of the exercises are too hard. There should always be a couple of easy ones, but I don't know this audience very well.