Corrigendum to “Intersection homology with field coefficients: K-Witt spaces and K-Witt bordism”

This note corrects an error in the char(K) = 2 case of the author’s computation of the bordism groups of K-Witt spaces for the field K. A complete computation is provided for the unoriented bordism group. For the oriented bordism groups, a nearly complete computation is provided as well a discussion of the difficulty of resolving a remaining ambiguity in dimensions equivalent to 2 mod 4. Recall that in [1], an n-dimensional K-Witt space, for a field K, is defined to be an oriented compact irreducible n-dimensional PL stratified pseudomanifold X satisfying the K-Witt condition that the lower-middle perversity intersection homology group IHk(L;K) is 0 for each link L of each stratum of X of dimension n − 2k − 1, k > 0. Following the definition of stratified pseudomanifold in [2], X does not possess codimension one strata, and “irreducible” means that X − Xn−2 is connected. Orientability is determined by the orientability of the top (regular) stratum. This definition generalizes Siegel’s definition in [10] of Q-Witt spaces (called there simply “Witt spaces”). The motivation for this definition is that such spaces possess intersection homology Poincaré duality IHk(X;K) ∼= Hom(IHn−k(X;K), K). The author’s paper [1] concerns K-Witt spaces and, in particular, a computation of the bordism theory of such spaces ΩK−Witt ∗ . However, there is an error in the computation of the coefficient groups ΩK−Witt 4k+2 when char(K) = 2. It is claimed in [1] that ΩK−Witt 4k+2 = 0. When char(K) > 2, the null-bordism of a 4k + 2 dimensional K-Witt space X is established in [1] by following Siegel’s computation [10] for Q-Witt spaces by performing a sequence of singular surgeries to obtain a space X ′ such that IH2k+1(X ′;K) = 0. The K-Witt null-bordism of X is the union of the trace of the surgeries from X to X ′ with the closed cone c̄X ′. One performs such surgeries on elements [z] ∈ IH2k+1(X;K) such that the Goresky-MacPherson intersection product [z] · [z] = 0. As the intersection product is skew symmetric on IH2k+1(X;K), such a [z] always exists. The error in [1] stems from overlooking the fact that this last fact is not necessarily true in 2000 Mathematics Subject Classification: 55N33, 57Q20, 57N80