Generating the homology of covers of surfaces

被引:0
|
作者
Boggi, Marco [1 ]
Putman, Andrew [2 ,3 ]
Salter, Nick [2 ]
机构
[1] Univ Fed Fluminense, Inst Matemat Estat, Niteroi, RJ, Brazil
[2] Univ Notre Dame, Dept Math, Notre Dame, IN USA
[3] Univ Notre Dame, Dept Math, 255 Hurley Hall, Notre Dame, IN 46556 USA
关键词
FINITE COVERS; QUOTIENTS; SUBGROUPS; CURVES;
D O I
10.1112/blms.13026
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Putman and Wieland conjectured that if sigma similar to ->sigma$\widetilde{\Sigma }\rightarrow \Sigma$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ under the action of lifts to sigma similar to$\widetilde{\Sigma }$ of mapping classes on sigma$\Sigma$ are infinite. We prove that this holds if H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ is generated by the homology classes of lifts of simple closed curves on sigma$\Sigma$. We also prove that the subspace of H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that H1(sigma similar to;Q)$\operatorname{H}_1(\widetilde{\Sigma };\mathbb {Q})$ is generated by the homology classes of lifts of loops on sigma$\Sigma$ lying on subsurfaces homeomorphic to 3-holed spheres.
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页码:1768 / 1787
页数:20
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