Homotopy groups of ascending unions of infinite-dimensional manifolds

TL;DR

This paper establishes conditions under which the homotopy groups of an infinite-dimensional manifold formed as an ascending union are the direct limits of the homotopy groups of the constituent manifolds, aiding Lie theory applications.

Contribution

It provides a framework to compute homotopy groups of infinite-dimensional manifolds as direct limits, extending previous results to dense and uncountable unions.

Findings

Homotopy groups of the union equal the direct limit of the homotopy groups of the steps.

Results apply to dense unions and uncountable directed unions.

Facilitates understanding of Lie group extensions via homotopy groups.

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.