[Paper Review] The homotopy groups of the inverse limit of a tower of fibrations

Philip Hirschhorn|arXiv (Cornell University)|Jul 6, 2015

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper provides a rigorous, elementary proof of the classical result that the homotopy groups of the inverse limit of a tower of fibrations fit into a natural short exact sequence, with the kernel isomorphic to the lim¹ of the homotopy groups of one higher dimension. The key contribution is a detailed treatment of lim¹ for non-abelian groups, extending the classical result beyond simply connected or abelian cases.

ABSTRACT

We carefully present an elementary proof of the well known theorem that each homotopy group (or, in degree zero, pointed set) of the inverse limit of a tower of fibrations maps naturally onto the inverse limit of the homotopy groups (or, in degree zero, pointed sets) of the spaces in the tower, with kernel naturally isomorphic to $\lim^{1}$ of the tower of homotopy groups of one dimension higher.

Motivation & Objective

• To provide a self-contained, elementary proof of the homotopy exact sequence for inverse limits of fibrations.
• To clarify the role of lim¹ in the long exact sequence of homotopy groups for inverse limits.
• To extend the classical result beyond simply connected or abelian fundamental groups, covering general pointed spaces with fibrations.
• To establish the naturality and exactness of the sequence in all degrees, including degree zero as a pointed set.
• To prove that weak equivalences on each level of a tower of fibrations induce weak equivalences on the inverse limits.

Proposed method

• Define lim¹ for a tower of groups via a group action on the product set, with orbits forming the pointed set lim¹.
• Construct a natural map from πₖ(lim Xₙ) to lim πₖXₙ and analyze its kernel.
• Show that the kernel is isomorphic to lim¹ πₖ₊₁Xₙ using explicit homotopy lifting and path-lifting techniques.
• Use induction and homotopy extension to construct compatible homotopies across the tower, ensuring compatibility with the inverse limit structure.
• Prove surjectivity of the map P: πₖ(lim Xₙ) → lim πₖXₙ via path-lifting and inductive construction of compatible representatives.
• Establish that the kernel map is a homomorphism when all Xₙ are H-spaces and maps are H-maps, using the H-space product structure.

Experimental results

Research questions

• RQ1 How can the classical lim¹ exact sequence for homotopy groups of inverse limits be proven in full generality, including non-abelian and non-simply-connected cases?
• RQ2 What is the precise algebraic structure of lim¹ for towers of non-abelian groups, and how does it relate to homotopy groups?
• RQ3 Under what conditions is the natural map πₖ(lim Xₙ) → lim πₖXₙ surjective with kernel isomorphic to lim¹ πₖ₊₁Xₙ?
• RQ4 How does the H-space structure on the spaces in the tower affect the group structure of the kernel?
• RQ5 What conditions ensure that a weak equivalence on each level of a tower of fibrations induces a weak equivalence on the inverse limit?

Key findings

• The homotopy groups of the inverse limit of a tower of fibrations fit into a natural short exact sequence: 1 → lim¹ πₖ₊₁Xₙ → πₖ(lim Xₙ) → lim πₖXₙ → 1.
• For k ≥ 2, this is a short exact sequence of abelian groups; for k = 1, it is a sequence of groups; for k = 0, it is a sequence of pointed sets.
• The kernel of the map πₖ(lim Xₙ) → lim πₖXₙ is naturally isomorphic to lim¹ πₖ₊₁Xₙ, with the action defined via the tower of fibrations.
• When all Xₙ are H-spaces and all structure maps are H-maps, the kernel lim¹ π₁Xₙ inherits a group structure and the kernel map is a homomorphism.
• A map of towers of fibrations that is a weak equivalence at each level induces a weak equivalence on the inverse limits, as shown via the five lemma and path-lifting arguments.
• The proof establishes that the map P: πₖ(lim Xₙ) → lim πₖXₙ is surjective, and the kernel is precisely lim¹ πₖ₊₁Xₙ via explicit construction of compatible homotopies.