Putman and Wieland conjectured that if is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of under the action of lifts to of mapping classes on are infinite. We prove that this holds if is generated by the homology classes of lifts of simple closed curves on . We also prove that the subspace of spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that is generated by the homology classes of lifts of loops on lying on subsurfaces homeomorphic to 3-holed spheres.

中文翻译：

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生成表面覆盖的同源性

普特曼和维兰德推测，如果是足够高亏格的闭合定向表面之间的有限分支覆盖，则所有非零元素的轨道在升降机的作用下的映射类是无限的。我们证明这成立，如果由简单闭合曲线的升力的同源类生成。我们还证明了子空间由这样的电梯跨越的是一个辛子空间。最后，简单的闭合曲线位于与 2 孔球体同胚的次曲面上，我们证明由循环提升的同源类生成位于与三孔球体同胚的次表面上。