infinite-dimensional流形升并的同伦群
Homotopy groups of ascending unions of infinite-dimensional manifolds
摘要：
设M是拓扑向量空间上的一个拓扑流形，它是这类流形M_n的一个上升序列的并.对每个非负整数k，我们给出了一个温和的条件，使得M的第k个同伦群是M_n的第k个同伦群的直接极限.这个结果对于李群理论是有用的，因为无穷维李群G的许多重要例子可以表示为有限维或无穷维李群的升并（它们的同伦群可能更容易获得）。对于k=0，k=1和k=2，需要关于G的第k个同伦群的信息来理解具有交换核的G的李群扩张。当阶数M_n的并在M中仅稠密时（在适当的假设下），上述结论仍然成立。此外，升序联合可以被（可能是不可数的）有向联合所取代。
英文摘要：
Let M be a topological manifold modelled on topological vector spaces, which is the union of an ascending sequence of such manifolds M_n. We formulate a mild condition ensuring that the k-th homotopy group of M is the direct limit of the k-th homotopy groups of the steps M_n, for each non-negative integer k. This result is useful for Lie theory, because many important examples of infinite-dimensional Lie groups G can be expressed as ascending unions of finite- or infinite-dimensional Lie groups (whose homotopy groups may be easier to access). Information on the k-th homotopy groups of G, for k=0, k=1 and k=2, is needed to understand the Lie group extensions of G with abelian kernels. The above conclusion remains valid if the union of the steps M_n is merely dense in M (under suitable hypotheses). Also, ascending unions can be replaced by (possibly uncountable) directed unions.
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