第三上同调与李群上的堆叠
Third Group Cohomology and Gerbes over Lie Groups
AI总结：
本文研究了李群上堆叠的第三上同调与局部光滑上同调的关系，并探讨了在特定群扩张条件下构造转格映射的方法。
摘要：
堆叠作为复希尔伯特空间的项目单位群的主丛，在拓扑空间H上的拓扑分类由第三上同调H³(H, Z)给出。当H为拓扑群时，整数上同调通常与H的局部连续（或在李群情况下局部光滑）第三群上同调相关。本文将详细研究这种关系，特别是在群扩张1→N→G→H→1的情况下，当堆叠由N的阿贝尔扩张1→A→N̂→N→1定义时。特别是当H₁_s(N,A)消失时，将构造一个转格映射H²_s(N, A)→H³_s(H, A^N)，其中A^N是A中的N不变子群，下标s表示局部光滑上同调。这种关系的例子出现在规范理论中，并在本文中讨论。
英文摘要：
The topological classification of gerbes, as principal bundles with the structure group the projective unitary group of a complex Hilbert space, over a topological space $H$ is given by the third cohomology $\text{H}^3(H, \Bbb Z)$. When $H$ is a topological group the integral cohomology is often related to a locally continuous (or in the case of a Lie group, locally smooth) third group cohomology of $H$. We shall study in more detail this relation in the case of a group extension $1\to N \to G \to H \to 1$ when the gerbe is defined by an abelian extension $1\to A \to \hat N \to N \to 1$ of $N$. In particular, when $\text{H}_s^1(N,A)$ vanishes we shall construct a transgression map $\text{H}^2_s(N, A) \to \text{H}^3_s(H, A^N)$, where $A^N$ is the subgroup of $N$-invariants in $A$ and the subscript $s$ denotes the locally smooth cohomology. Examples of this relation appear in gauge theory which are discussed in the paper.
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