论文标题
Instantons和理性同源性领域
Instantons and rational homology spheres
论文作者
论文摘要
在先前的工作中，第二作者定义了“ iPovariant Instanton同源群”的$ i^\ bullet（y，π; r）$，用于有理同源性3-Sphere $ y $，一组辅助数据$π$和PID $ r $。这些对象是同时戒指$ h^{ - *}（bso_3; r）$的模块。 我们证明，Equivariant Instanton同源性组$ i^\ bullet（y; r）$独立于辅助数据$π$，因此定义了理性同源性领域的拓扑不变。此外，我们证明这些不变性在3个manifolds的共同体下起作用，并在边界组件之间具有路径。 对于任何理性同源性球体$ y $，我们还可以定义Floer不可约的Instanton同源性集体组的类似物，与Equivariant Instanton同源组不同。 However, our methods allow us to prove a precise "wall-crossing formula'' for $I_*(Y, π; R)$ as the auxiliary data $π$ moves between adjacent chambers. We use this to define an instanton invariant $λ_I(Y) \in \Bbb Q$ of rational homology spheres, conjecturally equal to the Casson-Walker invariant. 我们的不变性方法使用一种新型技术，称为悬浮流类别。鉴于障碍物的cobordism $ w：y \ to y'$，它支持可简化的intsantons，既不能横向切割，也不能通过对扰动的微小变化来删除，我们删除并替换了Instantons模量空间中阻塞的解决方案的邻域。所得的模量空间具有新型的边界组件，因此请勿在$ y $和$ y'$的intsanton链络合物之间定义链条图。但是，它确实定义了$ y $的Instanton链综合体和$ Y'$的Instanton Chain Complex的一种悬架之间的链条图。
In previous work, the second author defined 'equivariant instanton homology groups' $I^\bullet(Y,π;R)$ for a rational homology 3-sphere $Y$, a set of auxiliary data $π$, and a PID $R$. These objects are modules over the cohomology ring $H^{-*}(BSO_3;R)$. We prove that the equivariant instanton homology groups $I^\bullet(Y;R)$ are independent of the auxiliary data $π$, and thus define topological invariants of rational homology spheres. Further, we prove that these invariants are functorial under cobordisms of 3-manifolds with a path between the boundary components. For any rational homology sphere $Y$, we may also define an analogue of Floer's irreducible instanton homology group of integer homology spheres $I_*(Y, π; R)$ which now depends on the auxiliary data $π$, unlike the equivariant instanton homology groups. However, our methods allow us to prove a precise "wall-crossing formula'' for $I_*(Y, π; R)$ as the auxiliary data $π$ moves between adjacent chambers. We use this to define an instanton invariant $λ_I(Y) \in \Bbb Q$ of rational homology spheres, conjecturally equal to the Casson-Walker invariant. Our approach to invariance uses a novel technique known as a suspended flow category. Given an obstructed cobordism $W: Y \to Y'$, which supports reducible instantons which can neither be cut out transversely nor be removed by a small change to the perturbation, we remove and replace a neighborhood of obstructed solutions in the moduli space of instantons. The resulting moduli spaces have a new type of boundary component, so do not define a chain map between the instanton chain complexes of $Y$ and $Y'$. However, it does define a chain map between the instanton chain complex of $Y$ and a sort of suspension of the instanton chain complex of $Y'$.
