I don't know whether this is a SES that every mathematician should know but it does satisfy the first sentence of the body of your question, since one could say it captures triangulability:

$$ 0 \to \text{ker}f \to \Theta_{3}^{H} \overset{f}{\to} \mathbb{Z}/2 \to 0 $$

where:

• the abelian group $\Theta_3^H$ is the cobordism group of oriented homology three spheres modulo binding an acylic PL/smooth 4-manifold.
• f is the Rokhlin homomorphism, which is 1/8th the signature of a compact, smooth spin(4) manifold that the integral homology sphere bounds.

Galewski, Stern and Matumoto showed in the 1980s that the non-splitting of this SES is equivalent to there being non-triangulable manifolds in every dimension 5 and above. Whereas, Manolescu recently showed that the SES does not in fact split.