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Homotopy groups of ascending unions of infinite-dimensional manifolds

By Helge Glöckner


Let M be a topological manifold modelled on topological vector spaces, which is the union of an ascending sequence M1 ⊆ M2 ⊆ · · · of such manifolds. We formulate a mild condition ensuring that πk(M,p) = lim πk(Mn,p) for all k ∈ N0 and p ∈ M. This result is useful for Lie theory, because many important examples of infinite-dimensional Lie groups can be expressed as ascending unions of finite- or infinitedimensional Lie groups (whose homotopy groups may be easier to access). Information on π0(G) = G/G0, π1(G) and π2(G) is needed to understand the Lie group extensions 1 → A → ̂ G → G → 1 of G with abelian kernels. The above conclusion remains valid if ⋃ n∈N Mn is merely dense in M (under suitable hypotheses). Also, ascending unions can be replaced by (possibly uncountable) directed unions

Topics: Contents
Year: 2008
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