Hatcher-Algebraic-Topology · Problem 2.1.30

Homology

Mathematics — Graph Theory · commutative-diagrams

Problem

In each of the following commutative diagrams assume that all maps but one are isomorphisms. Show that the remaining map must be an isomorphism as well. A⟶BA⟶BA⟶B↘  ↙↓    ↓↓    ↑CC⟶DC⟶D\begin{array}{ccc} A \longrightarrow B & \quad A \longrightarrow B & \quad A \longrightarrow B \\ \searrow\; \swarrow & \quad \downarrow\;\;\downarrow & \quad \downarrow\;\;\uparrow \\ C & \quad C \longrightarrow D & \quad C \longrightarrow D \end{array}
Intuitive explanation

The key insight is that commutativity lets you "transfer" invertibility along paths through the diagram. When all but one map are isomorphisms, you can compose and cancel them strategically to show the remaining map must also be bijective. For each diagram, identify which map is unknown, then use the fact that isomorphisms compose to isomorphisms and that commutativity gives you multiple routes between objects—if all other routes are invertible, the unknown path must be too.

Intuitive explanation

The key insight is that commutativity lets you "transfer" invertibility along paths through the diagram. When all but one map are isomorphisms, you can compose and cancel them strategically to show the remaining map must also be bijective. For each diagram, identify which map is unknown, then use the fact that isomorphisms compose to isomorphisms and that commutativity gives you multiple routes between objects—if all other routes are invertible, the unknown path must be too.

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