论文标题
$ eg_n $的cotangent束
Equivariant Lagrangian Floer homology via cotangent bundles of $EG_N$
论文作者
论文摘要
我们提供了Eprivariant Lagrangian Floer同源性$ HF_G（L_0，L_1）$的构造，用于紧凑型Lie Group $ G $，以hamiltonian的方式作用于Simplectic歧管$ M $，以及一对$ g $ g $ -Lagrangian submanifolds $ l_1 $ l_1 $ l_1 \ subset M $。 我们通过使用涉及$ EG $的近似值捆绑包的符号同质套件来做到这一点。我们的建筑依赖于Wehrheim和Woodward的被子理论以及望远镜的建设。 我们表明，这些群体在涉及建筑物的辅助选择中是独立的，并且是$ h^*（bg）$ - bimodules。在$ l_0 = l_1 $的情况下，我们表明他们的链复合物$ cf_g（l_0，l_1）$是同质的，等于$ l_0 $的莫尔斯摩尔斯群岛。 此外，如果零是当时映射$μ$的常规价值，并且如果$ g $在$μ^{ - 1}（0）$上自由起作用，我们将从$ cf_g（l_0，l_1）$构造两个“ kirwan morphisms” $ $ CF_G（L_0，L_1）$）。 我们的构造适用于精确和单调的设置，以及在Manolescu和Woodward的作品中考虑的Riemann Surface的Flat $ SU（2）$的扩展模量空间的环境中。应用于后一种设置，我们的构造为Atiyah-lloer猜想提供了一个e夫的符号侧。
We provide a construction of equivariant Lagrangian Floer homology $HF_G(L_0, L_1)$, for a compact Lie group $G$ acting on a symplectic manifold $M$ in a Hamiltonian fashion, and a pair of $G$-Lagrangian submanifolds $L_0, L_1 \subset M$. We do so by using symplectic homotopy quotients involving cotangent bundles of an approximation of $EG$. Our construction relies on Wehrheim and Woodward's theory of quilts, and the telescope construction. We show that these groups are independent in the auxiliary choices involved in their construction, and are $H^*(BG)$-bimodules. In the case when $L_0 = L_1$, we show that their chain complex $CF_G(L_0, L_1)$ is homotopy equivalent to the equivariant Morse complex of $L_0$. Furthermore, if zero is a regular value of the moment map $μ$ and if $G$ acts freely on $μ^{-1}(0)$, we construct two "Kirwan morphisms" from $CF_G(L_0, L_1)$ to $CF(L_0/G, L_1/G)$ (respectively from $CF(L_0/G, L_1/G)$ to $CF_G(L_0, L_1)$). Our construction applies to the exact and monotone settings, as well as in the setting of the extended moduli space of flat $SU(2)$-connections of a Riemann surface, considered in Manolescu and Woodward's work. Applied to the latter setting, our construction provides an equivariant symplectic side for the Atiyah-Floer conjecture.
