in the A Primer on Mapping Class Groups we have :
Geometric intersection number is a useful invariant but, as we will see, it is more difficult to compute than the algebraic intersection number.
what is mean of invariant ? why this is useful ?
is this mean homeomorphisms preserve geometric intersection number? or Geometric intersection number is invariant under homeomorphisms ? ( also algebraic intersection number invariant under homeomorphisms orientation preserving. ) let $h$ be a homeomorphism then $i(h(a),h(b))=i(a,b) $ ?
i know in mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects.but for Geometric intersection number where is this invariant ?
in here (Subgroups of Teichmuller Modular Groups By Nikolai V. Ivanov page 21) we have :
geometric intersection number of circles, it follows that it is invariant under the action of the modular group: $$i(h(a),h(b))=i(a,b) $$ for $h \in Mod(S)$