The Torelli geometry and its applications
The Torelli geometry and its applications
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DOI：
10.4310/mrl.2005.v12.n3.a2
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发表时间：
2003-11
影响因子：
1
通讯作者：
Benson Farb;N. V. Ivanov
中科院分区：
文献类型：
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作者：
Benson Farb;N. V. Ivanov
让年代是一个封闭的表面可定向的映射类集团属g。国防部(S) (S)被定义为orientationpreserving解析的年代的组合痕类→美国我们也需要扩展映射类组Mod±S (S)被定义为群合痕类的所有解析的年代→S .让我们修复美国的取向代数交叉数提供了非简并,斜对称的,双线性形式在H = H1(年代,Z),称为交叉形式。Mod(S)对H的自然作用保留了交点形式。如果我们在H上固定一个辛基，那么我们就可以用积分辛群Sp（2g，Z）来识别H的辛自同构群，并且modds对H的作用导致一个自然满射同态
Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod±(S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides a nondegenerate, skew-symmetric, bilinear form on H = H1(S,Z), called the intersection form. The natural action of Mod(S) on H preserves the intersection form. If we fix a symplectic basis in H, then we can identify the group of symplectic automorphisms of H with the integral symplectic group Sp(2g,Z) and the action of ModS on H leads to a natural surjective homomorphism