The decategorification of sutured Floer homology
The decategorification of sutured Floer homology
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DOI：
10.1112/jtopol/jtr007
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发表时间：
2011-01-01
影响因子：
1.1
通讯作者：
Rasmussen, Jacob
中科院分区：
文献类型：
--
作者：
Friedl, Stefan;Juhasz, Andras;Rasmussen, Jacob
对每一个平衡缝合流形（M，gamma）定义了一个挠不变量T，并证明了它与缝合Floer同调（SFH）的Euler特征一致.不变量T很容易使用Fox演算计算。借助于T，我们证明了：如果（M，gamma）与交错纽结的Seifert曲面互补，则在每个Spin（c）结构中，SFH（M，gamma）要么为0，要么为Z.挠不变量T也可以用来证明缝合流形是不可圆分解的，并区分Seifert曲面。SFH的支集在H（2）（M，偏导数M; R）上产生一个范数z。不变量T给出了范数z的下界，而范数z至多是缝合的瑟斯顿范数x（s）。对于闭3-流形，众所周知，Floer同调决定Thurston范数，但我们证明了z < x（s）可以在一般情况下发生。最后，我们计算T的几个广泛的缝合流形类。
We define a torsion invariant T for every balanced sutured manifold (M, gamma), and show that it agrees with the Euler characteristic of sutured Floer homology (SFH). The invariant T is easily computed using Fox calculus. With the help of T, we prove that if (M, gamma) is complementary to a Seifert surface of an alternating knot, then SFH(M, gamma) is either 0 or Z in every Spin(c) structure. The torsion invariant T can also be used to show that a sutured manifold is not disc decomposable, and to distinguish between Seifert surfaces.The support of SFH gives rise to a norm z on H(2)(M, partial derivative M; R). The invariant T gives a lower bound on the norm z, which in turn is at most the sutured Thurston norm x(s). For closed 3-manifolds, it is well known that Floer homology determines the Thurston norm, but we show that z < x(s) can happen in general. Finally, we compute T for several wide classes of sutured manifolds.