Homotopy groups of ascending unions of infinite-dimensional manifolds

Abstract

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.