论文标题
在第二个同置通勤元素的均匀空间组中
On the second homotopy group of spaces of commuting elements in Lie groups
论文作者
论文摘要
让$ g $为紧凑的连接谎言组，$ n \ geqslant 1 $整数。考虑在$ g $，$ hom（\ mathbb {z}^n，g）$中以$ n $ tumples的订购空间，并在伴随动作下的商，$ rep（\ mathbb {z}^n，g）：= home（\ hom（\ mathbb {z}^n，g）/g $。在本文中，我们研究并在许多情况下计算同型组$π_2（hOM（\ Mathbb {z}^n，g））$。对于$ g $简单 - 连接而简单，我们证明$π_2（\ hom（\ mathbb {z}^2，g））\ cong \ mathbb {z} $ and $π_2（pop（\ mathbb {z}^2，g）） $ hom（\ mathbb {z}^2，g）\ to rep（\ mathbb {z}^2，g）$诱导$ g $的dynkin索引乘以乘法。更普遍地表明，如果$ g $很简单，而$ hom（\ mathbb {z}^2，g）_ {1} \ subseteq hom（\ mathbb {z}^2，g）$是琐碎的同源物的路径，那么$ h_2（hOM（\ MATHBB {Z}^2，G）_ {1}; \ MATHBB {Z}）$是$π_1（g）^2 $ by $ \ MATHBB {Z} $的Schur乘数的扩展。 We apply our computations to prove that if $B_{com}G_{1}$ is the classifying space for commutativity at the identity component, then $π_4(B_{com}G_{1})\cong \mathbb{Z}\oplus \mathbb{Z}$, and we construct examples of non-trivial transitionally commutative structures on the琐事principal $ g $ -Bundle覆盖了球体$ \ Mathbb {s}^{4} $。
Let $G$ be a compact connected Lie group and $n\geqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(\mathbb{Z}^n,G)$, and its quotient under the adjoint action, $Rep(\mathbb{Z}^n,G):=Hom(\mathbb{Z}^n,G)/G$. In this article we study and in many cases compute the homotopy groups $π_2(Hom(\mathbb{Z}^n,G))$. For $G$ simply--connected and simple we show that $π_2(Hom(\mathbb{Z}^2,G))\cong \mathbb{Z}$ and $π_2(Rep(\mathbb{Z}^2,G))\cong \mathbb{Z}$, and that on these groups the quotient map $Hom(\mathbb{Z}^2,G)\to Rep(\mathbb{Z}^2,G)$ induces multiplication by the Dynkin index of $G$. More generally we show that if $G$ is simple and $Hom(\mathbb{Z}^2,G)_{1}\subseteq Hom(\mathbb{Z}^2,G)$ is the path--component of the trivial homomorphism, then $H_2(Hom(\mathbb{Z}^2,G)_{1};\mathbb{Z})$ is an extension of the Schur multiplier of $π_1(G)^2$ by $\mathbb{Z}$. We apply our computations to prove that if $B_{com}G_{1}$ is the classifying space for commutativity at the identity component, then $π_4(B_{com}G_{1})\cong \mathbb{Z}\oplus \mathbb{Z}$, and we construct examples of non-trivial transitionally commutative structures on the trivial principal $G$-bundle over the sphere $\mathbb{S}^{4}$.
