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We investigate the relation between the topological entropy of pseudo-Anosovmaps on surfaces with punctures and the rank of the first homology of theirmapping tori. On the surface $S$ of genus $g$ with $n$ punctures, we show thatthe entropy of a pseudo-Anosov map is bounded from above by$\dfrac{(k+1)\log(k+3)}{|\chi(S)|}$ up to a constant multiple when the rank ofthe first homology of the mapping torus is $k+1$ and $k, g, n$ satisfy acertain assumption. This is a partial generalization of precedent works of Tsaiand Agol-Leininger-Margalit.

Topological entropy of pseudo-Anosov maps on punctured surfaces vs. homology of mapping tori