On a genus-two surface, let $\alpha$ be a separating simple closed curve that cuts the surface into two once-punctured tori. Choose a simple closed curve $\beta$ that passes from one side to the other and back, with the two crossings arranged in minimal position. The crossings have opposite signs, so
$$
\langle\alpha,\beta\rangle=0,
$$
but the <bigon criterion> shows that they cannot be removed and hence
$$
i(\alpha,\beta)=2.
$$
Equivalently, $\alpha$ is separating and therefore represents zero in <first homology>, forcing its algebraic intersection with every curve to vanish even though its geometric intersection need not vanish.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 3 / d / Solution

{title2=$H_1(X)$}

The first homology group $H_1(X)$ is the group of one-dimensional cycles modulo boundaries. For a path-connected space it is the <abelianization> of the <fundamental group>.

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