论文标题
单极浮子同源性和不变的theta特性
Monopole Floer homology and invariant theta characteristics
论文作者
论文摘要
我们描述了三序的单极浮子同源性与里曼表面的几何形状之间的关系。考虑一个紧凑的Riemann Surface $σ$的自动形态$φ$，带有$ \ Mathbb {p}^1 $。 $σ$上的theta特性$ l $之间存在自然对应关系，$σ$下的$φ$下是不变的，自轭旋转$^c $ structures $ \ mathfrak {s} _l $上的映射torus $m_φ$的$ $ $φ$。我们表明，$（m_φ，\ mathfrak {s} _l）$的单极浮子同源性组明确地由$ h^0（l）$的$φ$（提升）的特征值确定，这是$ l $ $ l $的holomorphic部分的空间。对我们的计算进行了删除，我们还获得了$ h^0（l）$的尺寸等于$（m_φ，\ mathfrak {s} _l）$的reidemeister-turaev扭转。最后，我们将描述与Atiyah-Bott $ G $ -SPIN定理相结合，以在RAMIFIENT数据方面为所有自动形态的浮点同源组$φ$提供明确的计算。
We describe a relationship between the monopole Floer homology of three-manifolds and the geometry of Riemann surfaces. Consider an automorphism $φ$ of a compact Riemann surface $Σ$ with quotient $\mathbb{P}^1$. There is a natural correspondence between theta characteristics $L$ on $Σ$ which are invariant under $φ$ and self-conjugate spin$^c$ structures $\mathfrak{s}_L$ on the mapping torus $M_φ$ of $φ$. We show that the monopole Floer homology groups of $(M_φ,\mathfrak{s}_L)$ are explicitly determined by the eigenvalues of the (lift of the) action of $φ$ on $H^0(L)$, the space of holomorphic sections of $L$. Decategorifying our computation, we also obtain that the dimension of $H^0(L)$ equals the Reidemeister-Turaev torsion of $(M_φ,\mathfrak{s}_L)$. Finally, we combine our description with the Atiyah-Bott $G$-spin theorem to provide explicit computations of the Floer homology groups for all automorphisms $φ$ of prime order in terms of ramification data.
