Product formulae for Ozsváth-Szabó 4-manifold invariants
Product formulae for Ozsváth-Szabó 4-manifold invariants
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Ozsváth-Szabó 4 流形不变量的乘积公式
DOI：
10.2140/gt.2008.12.1557
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发表时间：
2007
影响因子：
2
通讯作者：
Thomas E. Mark
中科院分区：
文献类型：
--
作者：
Stanislav Jabuka;Thomas E. Mark
我们给出了4-流形X的Ozsvath-Szabo不变量的公式，这些不变量是由具有平凡法丛和亏格g 1的两个流形M_1、M_2沿曲面U_1、U_2的纤维和得到的。这些公式是由沿公共边界粘合两个4-流形的结果的Ozsvath-Szabo不变量的一个一般定理得到的，该不变量是用片断的相对不变量表示的。这些相对不变量取值于某些Novikov环上模的系数的Heegaard Floer同调形式；纤维和公式源于这样一个定理，即当所讨论的4-流形具有b-C2时，这种“扰动”版本的Heegaard-Floer理论恢复了通常的Ozsvath-Szabo不变量。这种构造允许将Ozsvath-Szabo不变量的定义扩展到具有b-Cd1的4-流形，这取决于某些选择，这与Seiberg-Witten理论非常相似。该乘积公式可以快速地计算各种4维流形的Ozsvath-Szabo不变量；在所有情况下，结果都符合Ozsvath-Szabo和Seiberg-Witten不变量之间的等价性猜想。
We give formulae for the Ozsvath‐Szabo invariants of 4‐manifolds X obtained by fiber sum of two manifolds M1 , M2 along surfaces U1 , U2 having trivial normal bundle and genus g 1. The formulae follow from a general theorem on the Ozsvath‐ Szabo invariants of the result of gluing two 4‐manifolds along a common boundary, which is phrased in terms of relative invariants of the pieces. These relative invariants take values in a version of Heegaard Floer homology with coefficients in modules over certain Novikov rings; the fiber sum formula follows from the theorem that this “perturbed” version of Heegaard Floer theory recovers the usual Ozsvath‐Szabo invariants, when the 4‐manifold in question has b C 2. The construction allows an extension of the definition of Ozsvath‐Szabo invariants to 4‐manifolds having b C D 1 depending on certain choices, in close analogy with Seiberg‐Witten theory. The product formulae lead quickly to calculations of the Ozsvath‐Szabo invariants of various 4‐manifolds; in all cases the results are in accord with the conjectured equivalence between Ozsvath‐Szabo and Seiberg‐Witten invariants.