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For a topological space $X$ and a subspace $A$, let $Q_n(X,A)$ be the group of singular cubical $n$-chains of $X$ relative to $A$ and let $D_n(X,A)$ be the subgroup of degenerate cubical chains. The homology of the quotient complex $Q_{\bullet}(X,A)/D_{\bullet}(X,A)$ is then isomorphic to ordinary singular homology (I think this is from Serre's thesis?).

However, as far as I can tell the homology groups of $Q_{\bullet}(X,A)$ satisfy all the axioms of a homology theory except the dimension axiom, where it is easy to see that $H_k(Q_{\bullet}(\text{pt},\emptyset))$ is $\mathbb{Z}$ for all $k \geq 0$.

Question: what homology theory is computed by $Q_{\bullet}(X,A)$?

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  • $\begingroup$ I would be surprised is this satisfied the homotopy axiom: two continuous maps $f,g:X\to Y$ that are homotopic induce the same map at the level of homology. $\endgroup$
    – André Henriques
    Commented Jan 8, 2023 at 22:12
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    $\begingroup$ No, I think the homotopy axiom is okay. And I'm sure that $H_kQ_\bullet$ is $\oplus_{n\ge 0} H_{k-n}$. It might be fun and instructive to think of a simple reason why. $\endgroup$
    – Tom Goodwillie
    Commented Jan 9, 2023 at 1:14

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