[Paper Review] The structure set of an arbitrary space, the algebraic surgery exact sequence and the total surgery obstruction

Andrew Ranicki|ArXiv.org|Nov 30, 2001

Homotopy and Cohomology in Algebraic Topology11 references3 citations

TL;DR

This paper establishes a complete algebraic surgery exact sequence for arbitrary Poincaré complexes, introducing the total surgery obstruction as a chain-level invariant that detects when such a complex is homotopy equivalent to a topological manifold. It identifies the structure set of a space with the relative surgery groups in the algebraic surgery exact sequence, proving that the total surgery obstruction vanishes if and only if the space is simple homotopy equivalent to a topological manifold (for dimension ≥ 5).

ABSTRACT

An introduction to the applications of algebraic surgery to the structure theory of high-dimensional topological manifolds.

Motivation & Objective

• To extend the algebraic surgery exact sequence to arbitrary spaces with n-dimensional Poincaré duality, not just manifolds.
• To define and characterize the total surgery obstruction s(X) as a chain-level invariant measuring the failure of a Poincaré complex to be homotopy equivalent to a topological manifold.
• To establish a precise correspondence between the geometric structure set of a manifold and the algebraic surgery exact sequence via the total surgery obstruction.
• To extend the framework to homology manifolds using a 4-periodic surgery exact sequence and relate the resolution obstruction to the total surgery obstruction.

Proposed method

• Define the (Z,X)-category of based finitely generated free Z-modules with X-local structure, providing the domain for the assembly map.
• Construct the assembly map A: H_n(X; L•) → L_n(Z[π₁(X)]) as a natural transformation from generalized homology with L-theory coefficients to surgery obstruction groups.
• Identify the relative surgery groups S_n(X) as the cobordism groups of quadratic Poincaré complexes over (Z,X) that assemble to contractible complexes over Z[π₁(X)].
• Define the total surgery obstruction s(X) ∈ S_n(X) as the cobordism class of a quadratic Poincaré complex over (Z,X) with contractible assembly over Z[π₁(X)].
• Use the algebraic surgery exact sequence to identify the Browder-Novikov-Sullivan-Wall surgery exact sequence for topological manifolds with the corresponding portion of the algebraic sequence.
• Extend the framework to homology manifolds via a 4-periodic surgery exact sequence involving the groups S̄_n(X) and the resolution obstruction i(X) ∈ H_n(X; L₀(Z)).

Experimental results

Research questions

• RQ1 What is the algebraic characterization of a space with n-dimensional Poincaré duality being homotopy equivalent to a topological manifold?
• RQ2 How can the structure set of a topological manifold be algebraically described using the algebraic surgery exact sequence?
• RQ3 What is the precise role of the total surgery obstruction s(X) in detecting the manifold structure of a Poincaré complex?
• RQ4 How does the algebraic surgery framework extend to homology manifolds, and what is the relationship between the resolution obstruction and the total surgery obstruction?
• RQ5 Under what conditions is the assembly map injective or an isomorphism, and how does this relate to the Borel rigidity and Novikov conjectures?

Key findings

• The total surgery obstruction s(X) ∈ S_n(X) vanishes if and only if an n-dimensional simple Poincaré complex X is simple homotopy equivalent to a topological manifold, provided n ≥ 5.
• The structure set S^TOP(M) of a topological n-manifold M (n ≥ 5) is identified with the group S_{n+1}(M) in the algebraic surgery exact sequence.
• The structure invariant s(h) ∈ S_{n+1}(M) of a simple homotopy equivalence h: N → M between n-manifolds vanishes if and only if h is homotopic to a homeomorphism.
• For homology manifolds, the total surgery obstruction s(X) is the image of the resolution obstruction i(X) ∈ H_n(X; L₀(Z)) under a natural map.
• The homology manifold surgery sequence for n ≥ 6 is in one-to-one correspondence with a portion of the 4-periodic algebraic surgery exact sequence.
• The structure set of S^n (n ≥ 6) as a homology manifold is isomorphic to L₀(Z), detected by the resolution obstruction.