A construction of surface bundles over surfaces with non-zero signature
A construction of surface bundles over surfaces with non-zero signature
复制标题
具有非零签名的表面上的表面束的构造
DOI：
10.18910/4596
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发表时间：
1998
影响因子：
0.4
通讯作者：
Hisaaki Endo
中科院分区：
文献类型：
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作者：
Hisaaki Endo
设f（g）/f（J）是亏格为g（h）的闭定向曲面，其中g（h）是非负整数.设Diff+ Diff h是具有C°°-拓扑的diff/j的所有保向微分同态的群。一个在一个曲面上的丛（也称为一个曲面丛）是在一个曲面上的纤维丛，其总空间为E，纤维丛为p，投影为p：E -> Diff g，结构群为Diff + Diff h。我们主要关心的是π的全空间E的签名τ（E）。很容易看出，如果ε是平凡丛，则r（E）= τ（ε g）r（ε h）- 0。Chern-Hirzebruch-Serre [5]证明了如果n_g的基本群7 r（n_5）平凡地作用在n_g的上同调环H*（n_h]R）上，则r（E）= 0。科代拉[121]和Atiyah [1]给出了具有非零签名的曲面上的曲面丛的例子。对于每对（m，t）整数m，t ∈ Z（m > 2，ε> 3），科代拉构造了一个曲面丛f = f（ra，t），其中
Let Σg (respectively Σ/J be a closed oriented surface of genus g (respectively h), where g (respectively h) is a non-negative integer. Let Diff+Σh be the group of all orientation-preserving difϊeomorphisms of Σ/j with C°°-topology. A Σ^bundle over Σg (also called a surface bundle over a surface) is fiber bundle ξ = (£", Σg,p, Σ/^Diff+Σfr) over Σ^ with total space E, fiber Σ&, projection p : E —> Σg and structure group Diff+Σh. Our main concern is the signature τ(E) of the total space E of ξ. It is easily seen that if ξ is a trivial bundle then r(E) = τ(Σg)r(Σh) — 0. ChernHirzebruch-Serre [5] proved that if the fundamental group 7r(Σ5) of Σg acts trivially on the cohomology ring H*(Σh]R) of Σ^ then r(E) = 0. Kodaira [121 and Atiyah [1] gave examples of surface bundles over surfaces with non-zero signature. For each pair (m,t) of integers m,t G Z(m > 2,£ > 3), Kodaira constructed a surface bundle ξ = f (ra, t) with