Quandle coverings and their Galois correspondence
Quandle coverings and their Galois correspondence
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Quandle 覆盖物及其伽罗瓦对应关系
DOI：
10.4064/fm225-1-7
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发表时间：
2006
影响因子：
0.6
通讯作者：
Michael Eisermann
中科院分区：
文献类型：
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作者：
Michael Eisermann
本文建立了Quandles的代数覆盖理论。对于每个连通Quandle，我们显式地构造了一个泛覆盖，进而将代数基本群定义为泛覆盖的自同构群。然后，我们建立了连通覆盖与基本群的子群之间的Galois对应。因此，Quandle覆盖在形式上类似于拓扑空间的覆盖，并且类似于Kervaire的完全群的代数覆盖理论。详细的调查也揭示了一些关键的差异，我们用大量的例子来说明这一点。作为应用，我们得到了Q的第二(上)同调群的一个简单公式.众所周知，H_1(Q)=H^1(Q)=\Z[pi_0(Q)].我们构造了自然同构H_2(Q)=\pi_1(q，q)_{ab}和H^2(q，A)=Ext(q，A)=Hom(\pi_1(q，q)，A)，这使人想起经典的1次Hurewicz同构.这意味着，当基本群已知时，(Co)二次同调计算变得非常容易。
This article establishes the algebraic covering theory of quandles. For every connected quandle we explicitly construct a universal covering, which in turn leads us to define the algebraic fundamental group as the automorphism group of the universal covering. We then establish the Galois correspondence between connected coverings and subgroups of the fundamental group. Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire's algebraic covering theory of perfect groups. A detailed investigation also reveals some crucial differences, which we illustrate by numerous examples. As an application we obtain a simple formula for the second (co)homology group of a quandle Q. It has long been known that H_1(Q) = H^1(Q) = \Z[\pi_0(Q)], and we construct natural isomorphisms H_2(Q) = \pi_1(Q,q)_{ab} and H^2(Q,A) = Ext(Q,A) = Hom(\pi_1(Q,q),A), reminiscent of the classical Hurewicz isomorphisms in degree 1. This means that whenever the fundamental group is known, (co)homology calculations in degree 2 become very easy.
DOI：
10.1007/978-1-4612-9839-7
发表时间：
1971
期刊：
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影响因子：
--
作者：
S. Lane
通讯作者：
S. Lane
Quandle 扭曲的亚历山大不变量和同调群
DOI：
--
发表时间：
2021
期刊：
影响因子：
--
作者：
Minami;R.;Yuta Taniguchi;谷口 雄大;谷口 雄大;谷口 雄大;谷口 雄大;谷口 雄大;Yuta Taniguchi;谷口 雄大;谷口雄大;Yuta Taniguchi;Yuta Taniguchi;谷口 雄大;谷口 雄大;谷口 雄大
通讯作者：
谷口 雄大