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In this paper, we introduce \textit{splitting numbers} of subvarieties in asmooth variety for a Galois cover, and prove that the splitting numbers areinvariant under certain homeomorphisms. By splitting numbers, we give anecessary and sufficient condition for two plane curves of type $(b,m)$ to betopologically equivalent as pairs of the complex projective plane and planecurves, where a plane curve of type $(b,m)$ is an arrangement of two smoothplane curves of degree $3$ and $b$ defined by I.~Shimada. Consequently, weprove that there are $\pi_1$-equivalent Zariski $k$-plets for any $k\geq2$.

A note on splitting numbers for Galois covers and $\pi_1$-equivalent Zariski $k$-plets