[Paper Review] Automorphisms of Manifolds and Algebraic K-Theory: Part III

Michael Weiss, Bruce Williams|arXiv (Cornell University)|Aug 19, 2013

Homotopy and Cohomology in Algebraic Topology62 references17 citations

TL;DR

This paper constructs a highly connected map from the structure space 𝒮(M) of a closed topological m-manifold M to a fiber of a local degree map on a spectrum combining algebraic L-theory and K-theory, refining surgery theory. The key result identifies the homotopy type of 𝒮(M) in the concordance stable range via a concoction of algebraic K- and L-theory, using assembly maps and controlled duality.

ABSTRACT

The structure space S(M) of a closed topological m-manifold M classifies bundles whose fibers are closed m-manifolds equipped with a homotopy equivalence to M. We construct a highly connected map from S(M) to a concoction of algebraic L-theory and algebraic K-theory spaces associated with M. The construction refines the well-known surgery theoretic analysis of the block structure space of M in terms of L-theory.

Motivation & Objective

• To compute the homotopy type of the structure space 𝒮(M) for a closed topological m-manifold M in the concordance stable range.
• To refine the classical surgery-theoretic description of the block structure space using algebraic K- and L-theory.
• To construct a homotopy invariant functor assigning to each space Y with a spherical fibration ξ a spectrum 𝐋𝐀_•_%(Y,ξ,m) combining L-theory and A-theory.
• To establish a highly connected map from 𝒮(M) to the fiber of the local degree map on this spectrum, particularly over 0 ∈ 8ℤ.
• To extend the framework to compact manifolds with nonempty boundary, maintaining formal similarity to the closed case.

Proposed method

• Define the spectrum Ω^{m}𝐋𝐀_•_%(Y,ξ,m) as the total homotopy fiber of a commutative square involving quadratic L-theory and Waldhausen A-theory with S¹∧ and homotopy orbit constructions.
• Use a category of finitely dominated retractive spaces or spectra over Y, equipped with Spanier-Whitehead duality relative to ξ and m.
• Employ the natural transformation Ξ from [37] to relate the rows of the diagram, enabling the construction of the algebraic approximation to the structure space.
• Construct the spectrum via homotopy fibers of homotopy pullbacks of rows in a diagram involving visible symmetric L-theory and A-theory with group actions.
• Apply a corrected Eilenberg swindle using endofunctors σ_i on a controlled algebraic K-theory category to show τ_* ≃ id + τ_*, implying τ_* is idempotent up to homotopy.
• Use the contractibility of U_∞(X) for Poincaré duality spaces X to establish existence and uniqueness of Spivak normal fibrations, crucial for the duality framework.

Experimental results

Research questions

• RQ1 How can the structure space 𝒮(M) of a closed topological m-manifold M be described in terms of algebraic K- and L-theory in the concordance stable range?
• RQ2 What is the precise homotopy-theoretic relationship between the structure space 𝒮(M) and a fiber of the local degree map on a combined K- and L-theory spectrum?
• RQ3 How does the inclusion of the homeomorphism group into the homotopy equivalence group relate to the structure space via homotopy fibers?
• RQ4 In what way does the algebraic approximation to the structure space refine classical surgery theory?
• RQ5 What role does the Spivak normal fibration play in ensuring the consistency of duality and assembly maps in the construction?

Key findings

• A highly connected map is constructed from the structure space 𝒮(M) to the fiber over 0 ∈ 8ℤ of the local degree map on the spectrum Ω^{∞+m}𝐋𝐀_•_%(M,ν,m), where ν is the normal fibration of M.
• The connectivity of this map is determined by the concordance stable range, which is defined precisely in Definition 11.5 and is approximately m/3 in practice.
• The spectrum 𝐋𝐀_•_%(Y,ξ,m) is defined as the total homotopy fiber of a commutative square involving quadratic L-theory and A-theory with S¹∧ and homotopy orbit functors.
• The construction is decoration-independent: any consistent choice of decoration from h, p, ..., ⟨−∞⟩ yields the same infinite loop space up to homotopy equivalence.
• The existence and uniqueness of Spivak normal fibrations for Poincaré duality spaces is established via the contractibility of U_∞(X), using regular neighborhoods in Euclidean space and embedded surgery.
• The Eilenberg swindle is applied via endofunctors σ_i on a controlled algebraic K-theory category, yielding τ_* ≃ id + τ_*, which implies τ_* is idempotent up to homotopy.