[Paper Review] Homotopy groups of ascending unions of infinite-dimensional manifolds

Helge Glöckner|arXiv (Cornell University)|Dec 26, 2008

Homotopy and Cohomology in Algebraic Topology40 references5 citations

TL;DR

This paper establishes conditions under which the homotopy groups of an ascending union of infinite-dimensional manifolds (or Lie groups) are isomorphic to the direct limit of the homotopy groups of the constituent manifolds. Using weak direct limit charts and compact regularity, it proves that πₖ(M,p) ≅ lim→ πₖ(Mₙ,p) for all k ∈ ℕ₀ and p ∈ M, enabling homotopy computation in infinite-dimensional Lie theory via finite-dimensional approximations.

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.

Motivation & Objective

• To establish sufficient conditions under which the homotopy groups of an ascending union of infinite-dimensional manifolds are isomorphic to the direct limit of the homotopy groups of the submanifolds.
• To extend Palais’ classical result on dense subspaces of locally convex spaces to non-linear settings involving manifolds and Lie groups.
• To provide a general framework for computing homotopy groups of infinite-dimensional Lie groups that are directed unions of finite- or infinite-dimensional Lie groups.
• To show that weak homotopy equivalence holds for inclusion maps from dense unions of Lie groups into their limits under suitable regularity conditions.
• To apply the results to concrete examples such as diffeomorphism groups and function spaces with compact or weighted smooth structures.

Proposed method

• Introduce the concept of a weak direct limit chart, which ensures compatibility between charts on the limit manifold and those on the submanifolds via continuous linear maps.
• Use the condition that each point in the manifold admits a weak direct limit chart to guarantee that the inclusion maps induce isomorphisms on all homotopy groups.
• Apply the notion of compact regularity (or compact retractivity) to ensure that compact sets in the limit manifold lie in some submanifold, enabling control over homotopy lifting.
• Leverage the direct limit structure of homotopy groups via the colimit construction lim→ πₖ(Mₙ,p) over the directed index set Aₚ = {α ∈ A : p ∈ Mₐ}.
• Verify that the conditions of the main theorem are satisfied in key examples, such as weighted diffeomorphism groups and smooth function spaces with compact support.
• Use the fact that if Mₙ ⊆ M are continuous inclusions and Mₙ are manifolds, then πₖ(M) ≅ lim→ πₖ(Mₙ) holds under weak direct limit chart and compact regularity assumptions.

Experimental results

Research questions

• RQ1 Under what conditions does the inclusion of an ascending union of infinite-dimensional manifolds into its limit induce isomorphisms on all homotopy groups?
• RQ2 Can Palais’ theorem on dense subspaces of locally convex spaces be generalized to non-linear manifolds and Lie groups?
• RQ3 How can homotopy groups of infinite-dimensional Lie groups be computed when they are expressed as directed unions of finite- or infinite-dimensional Lie groups?
• RQ4 What structural conditions (e.g., weak direct limit charts, compact regularity) ensure that the direct limit of homotopy groups of submanifolds recovers the homotopy groups of the full manifold?
• RQ5 To what extent does the inclusion map from a dense union of Lie groups into its limit manifold induce a weak homotopy equivalence?

Key findings

• If a manifold M is the directed union of submanifolds Mₐ with continuous inclusions and admits weak direct limit charts, then πₖ(M,p) ≅ lim→ πₖ(Mₐ,p) for all k ∈ ℕ₀ and p ∈ M.
• The result remains valid when the union ⋃ₐ Mₐ is dense in M, provided that M is compactly retractive and weak direct limit charts exist.
• The inclusion map from the dense union of compactly supported diffeomorphisms Diff_c(X) into the weighted diffeomorphism group Diff_𝒲(X)ᵇ is a weak homotopy equivalence.
• For a σ-compact smooth manifold M and a Lie group H, the inclusion C^∞_c(M,H) → C^r_c(M,H) is a weak homotopy equivalence for all r ∈ ℕ₀.
• The homotopy groups of the weighted diffeomorphism group Diff_𝒲(X)ᵇ are isomorphic to the direct limit of the homotopy groups of the compactly supported diffeomorphism groups Diff_K(X), i.e., πₖ(Diff_𝒲(X)ᵇ) ≅ lim→ πₖ(Diff_K(X)).
• The result applies to Lie groups modelled on (LF)-spaces or (DFS)-spaces, such as spaces of smooth vector fields or diffeomorphisms, under suitable regularity and approximation conditions.