Hatcher-Algebraic-Topology · Problem 2.1.26

This problem asks you to compare two homology groups built from the same topological data but in different ways, revealing that the quotient map doesn't preserve homological structure naively. The key insight is that $A$ is not a subcomplex—it's a closed subset with limit points—so the relative homology $H_1(X, A)$ and the reduced homology of the quotient $\widetilde{H}_1(X/A)$ measure fundamentally different things. Computing both groups explicitly will show they have different ranks or torsion, exposing how collapsing a non-subcomplex space can create or destroy cycles.

This problem asks you to compare two homology groups built from the same topological data but in different ways, revealing that the quotient map doesn't preserve homological structure naively. The key insight is that $A$ is not a subcomplex—it's a closed subset with limit points—so the relative homology $H_1(X, A)$ and the reduced homology of the quotient $\widetilde{H}_1(X/A)$ measure fundamentally different things. Computing both groups explicitly will show they have different ranks or torsion, exposing how collapsing a non-subcomplex space can create or destroy cycles.