[Paper Review] Witt groups of sheaves on topological spaces

Jonathan Woolf|ArXiv.org|Oct 10, 2005

Homotopy and Cohomology in Algebraic Topology13 references4 citations

TL;DR

This paper establishes that the Witt groups of cohomologically constructible sheaves on polyhedra form a generalized homology theory, identifying them with Ranicki's free symmetric L-groups when the coefficient ring is a regular Noetherian ring with 2 invertible. For rational coefficients, these groups are shown to be isomorphic to the 4-periodic colimit of bordism groups of PL Witt spaces, enabling the interpretation of L-classes of singular spaces as stable homology operations to rational homology.

ABSTRACT

This paper investigates the Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincare-Verdier duality. We consider two main cases, that of perfect complexes of sheaves on locally compact Hausdorff spaces and that of cohomologically constructible complexes of sheaves on polyhedra. We show that the Witt groups of the latter form a generalised homology theory for polyhedra and continuous maps. Under certain restrictions on the ring R, we identify the constructible Witt groups of a finite simplicial complex with Ranicki's free symmetric L-groups. Witt spaces are the natural class of spaces for which the rational intersection homology groups have Poincare duality. When the ring R is the rationals we show that every Witt space has a natural L-theory, or Witt, orientation and we identify the constructible Witt groups with the 4-periodic colimit of the bordism groups of Witt spaces. This allows us to interpret Goresky and Macpherson's L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology.

Motivation & Objective

• To develop a generalized homology theory for polyhedra using Witt groups of constructible sheaves.
• To identify the constructible Witt groups of finite simplicial complexes with Ranicki’s free symmetric L-groups under suitable ring conditions.
• To show that for rational coefficients, the constructible Witt groups coincide with the 4-periodic colimit of bordism groups of PL Witt spaces.
• To interpret L-classes of singular spaces as stable homology operations from constructible Witt homology to rational homology.
• To establish naturality and stability of L-classes under maps, products, and suspensions, using geometric and algebraic topology techniques.

Proposed method

• Define the Witt groups of triangulated categories of sheaves on topological spaces equipped with Poincaré–Verdier duality.
• Focus on perfect complexes on locally compact Hausdorff spaces and cohomologically constructible complexes on polyhedra.
• Use Balmer’s framework for Witt groups in triangulated categories with duality, requiring 2 to be invertible in the coefficient ring.
• Establish that constructible Witt groups satisfy the Eilenberg–Steenrod axioms, hence form a generalized homology theory.
• Apply Ranicki’s symmetric L-theory and algebraic bordism to identify the constructible Witt groups with $ H_*(K; bL^ullet(R)) $ for finite simplicial complexes $ K $.
• Use the 4-periodicity of L-theory and the signature of $ bCbP^2 $ to show stability of L-classes under product with $ bCbP^2 $, enabling the definition of total L-class on the colimit.

Experimental results

Research questions

• RQ1 Can the Witt groups of constructible sheaves on polyhedra be shown to form a generalized homology theory?
• RQ2 How do the constructible Witt groups of a finite simplicial complex relate to Ranicki’s free symmetric L-groups?
• RQ3 What is the relationship between the constructible Witt groups with rational coefficients and the bordism groups of PL Witt spaces?
• RQ4 How can L-classes of singular spaces be interpreted as stable homology operations from constructible Witt homology to rational homology?
• RQ5 What is the naturality and stability of the total L-class under maps, products, and suspensions in this context?

Key findings

• The constructible Witt groups of a finite simplicial complex $ K $ are isomorphic to Ranicki’s free symmetric L-groups $ H_*(K; bL^ullet(R)) $ when every finitely generated $ R $-module admits a finite free resolution.
• For $ R = bQ $, the constructible Witt groups are isomorphic to the 4-periodic colimit of the bordism groups of PL Witt spaces.
• The total L-class $ igoplus_i L_i: Omega^{ extrm{Witt}}_*(X,A) o H_*(X,A;bQ) $ is a natural transformation that commutes with maps, boundary maps, and products.
• The L-classes are stable under multiplication by $ bCbP^2 $, as $ L_0(bCbP^2) = 1 $, which allows the definition of the total L-class on the colimit of Witt homology.
• Torsion elements in the constructible Witt groups map to zero under the L-class, due to the rational homology being torsion-free.
• The L-classes of singular spaces are realized as stable homology operations from the constructible Witt homology to rational homology, providing a topological interpretation of L-theory invariants.