Witt groups of sheaves on topological spaces

Jon Woolf
2008 Commentarii Mathematici Helvetici  
This paper investigates the Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincaré-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
more » ... R, we identify these 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 Poincaré duality. When the ring R is the rationals we identify the constructible Witt groups with the 4-periodic colimit of the bordism groups of PL Witt spaces. This allows us to interpret L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology. Mathematics Subject Classification (2000 ). 32S60, 19G99, 55U30, 57Q20. Keywords. Witt groups, Witt spaces, intersection cohomology, L-theory, topology of singular spaces. 290 J. Woolf CMH simplicial complex K with Ranicki's free symmetric L-groups H * (K; L • (R)) [25, Proposition 14.5]. When R = Q 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 introduced in [28]. This answers Problem 6 in [10, §IX]. It also allows us to interpret L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology. Before giving further details we put these results into context. Witt groups and L-theory. In his 1937 paper [32] Witt studied symmetric bilinear forms over a field k, in particular defining what is now known as the Witt group W (k) -the set of isometry classes of symmetric bilinear forms (equipped with direct sum) modulo the stable equivalence relation generated by those forms with a Lagrangian subspace. (Witt also showed that the tensor product gives W (k) a natural ring structure, but we will ignore this for the present.) By analogy we can define the Witt group W (R) of any commutative ring R -see e.g. [22], [23]. This algebraic construction has been generalised to provide invariants in both algebraic geometry and in algebraic topology. In algebraic geometry, Knebusch defined the Witt group W (S) of a scheme S in [21] by considering symmetric bilinear forms on locally-free coherent sheaves (vector bundles) on S. In this context the classical Witt group W (R) of a ring R arises as W (Spec R). Knebusch's definitions can be used to define the Witt group of any exact category with duality. In a more recent development [3] Balmer extended this to define the Witt groups of any triangulated category T with duality. To obtain a good theory he requires that 2 be invertible, i.e., that the morphisms between any two objects are a Z[ 1 2 ]-module not merely an Abelian group. Balmer's Witt groups are a collection W i (T) of Abelian groups indexed by Z, but which turn out to be naturally 4-periodic, i.e., W i (T) ∼ = W i+4 (T). In a series of papers Balmer and others, notably Gille and Walter, have studied these groups for the derived category D lf (S) of locallyfree coherent sheaves on a scheme. Knebusch's Witt group W (S) is isomorphic to Balmer's zero'th Witt group W 0 (D lf (S)). Much of this work is summarised in [4, §5], which also contains a compendious bibliography. Of particular note is [18] in which it is shown that the Witt groups of the derived category of locally-free coherent sheaves on a regular scheme are representable in both the stable and unstable A 1 -homotopy categories. In algebraic topology, the development by Browder, Novikov, Sullivan and Wall of the surgery theory of high-dimensional manifolds in the 1960s culminated in the introduction by Wall [30] of the surgery obstruction groups L * (R). These L-groups are defined for any ring with involution R and are 4-periodic, i.e. L * (R) ∼ = L * +4 (R). Mishchenko and Ranicki also defined symmetric L-groups L * (R), with L 0 (R) = W (R) the classical Witt group. If 2 is invertible in R then L i (R) ∼ = L i (R), Vol. 83 (2008) Witt groups of sheaves on topological spaces J. Woolf CMH With the further assumption that 2 is invertible, we show that its Witt groups W p * (X) form a homotopy-invariant functor which satisfies all the axioms of a generalised homology theory apart from possibly excision. Let K be a simplicial complex. Its realisation is naturally stratified (see §4.2), and we denote this stratified space by K S . We can restrict our attention from the perfect complexes to the triangulated subcategory of complexes which are cohomologically constructible with respect to the stratification. This subcategory is also preserved by Poincaré-Verdier duality. Its Witt groups, which we dub the constructible Witt groups of K and denote W c * (K), form a generalised homology theory for simplicial complexes and simplicial maps. Simplicial approximation then allows us to obtain a generalised homology theory for compact polyhedra and continuous maps. The section ends with a brief discussion of equivariant generalisations and of related theories defined by altering the constructibility condition. In §4 we relate the constructible Witt groups W c * (K) of a finite simplicial complex K to Ranicki's free symmetric L-groups H * (K; L • (R)) by exhibiting a natural transformation from the latter to the former. If every finitely generated R-module has a finite resolution by finitely generated free R-modules then a theorem of Walter's [31, Theorem 5.3] shows that the natural transformation induces an isomorphism of point groups. Hence we obtain isomorphic generalised homology theories for simplicial complexes. Finally, §5 explains the geometric nature of the rational (R = Q) theory. We review Siegel's work [28] on the bordism groups of PL Witt spaces and construct a natural transformation from Witt bordism to the constructible Witt groups which is, in sufficiently high dimensions, an isomorphism. Phrased another way, the constructible Witt groups are the 4-periodic colimit of the Witt bordism groups. We use this geometric description, and an adaptation of the construction of L-classes in [24, §20], to view L-classes as homology operations from the bordism groups of Witt spaces, or, by the identification of the previous section, from the constructible Witt groups, to rational homology. Connections with other work. The isomorphism between certain constructible Witt groups and free symmetric L-groups constructed in §4 makes it apparent that this paper is closely related to Ranicki's work on L-theory. Our sheaf-theoretic approach has the virtues that it is technically simpler (at least for those familiar with sheaves and derived categories) and that it directly connects L-theory with the large body of work on intersection homology, self-dual complexes of sheaves and characteristic classes for singular spaces. This is not the first attempt to give a sheaf-theoretic description of L-theory. One could loosely describe Sections 3 and 4 as a triangulated version of Hutt's unpublished paper [20] , in which he considers the symmetric and quadratic L-groups of the category of complexes of sheaves with Poincaré-Verdier duality. However, there Vol. 83 (2008) Witt groups of sheaves on topological spaces
doi:10.4171/cmh/125 fatcat:3fbmveoub5bzvcszob6wgtnfwq