Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
Higher-order signature cocycles for subgroups of mapping class groups and homology cylinders
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映射类组和同源柱面子组的高阶签名共循环
DOI：
10.1093/imrn/rnr149
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发表时间：
2010
期刊：
影响因子：
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通讯作者：
Masaaki Suzuki
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文献类型：
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作者：
Masaaki Suzuki
我们为 S 的映射类组的元素定义不变量族，S 是一个紧凑的可定向表面。固定 pi_1(S) 的任何特征子组 H 并限制为 J(H)，即诱导恒等模 H 的映射类的任何子组。对于任何酉表示，pi_1(S)/H 的 r，我们将高阶 rho_r 不变量和签名 2-cocycle sigma_r 关联起来。这些签名余循环被证明是迈耶余循环的概括。特别地，每个 rho_r 是一个准同态，每个 sigma_r 是 J(H) 上的有界 2-cocycle。在最简单的非平凡情况之一中，通过改变 r，我们展示了无限族的线性无关拟同构和特征余循环。我们证明 rho_r 限制于某些有趣的子群上的同态。这些不变量中的许多自然地扩展到完整的映射类组，并且一些不变量扩展到基于 S 的同调柱面的幺半群。
We define families of invariants for elements of the mapping class group of S, a compact orientable surface. Fix any characteristic subgroup H of pi_1(S) and restrict to J(H), any subgroup of mapping classes that induce the identity modulo H. To any unitary representation, r of pi_1(S)/H we associate a higher-order rho_r-invariant and a signature 2-cocycle sigma_r. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular each rho_r is a quasimorphism and each sigma_r is a bounded 2-cocycle on J(H). In one of the simplest non-trivial cases, by varying r, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the rho_r restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on S.