K-Witt bordism in characteristic 2

This note provides a computation of the bordism groups of K-Witt spaces for fields K with characteristic 2. We provide a complete computation for the unoriented bordism groups. 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. This corrects an error in the char(K) = 2 case of the author’s prior computation of the bordism groups of K-Witt spaces for an arbitrary field K. In [1], an n-dimensional K-Witt space, for a field K, is defined to be an oriented compact 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. Orientability is determined by the orientability of the top (regular) strata. This definition generalizes Siegel’s definition in [11] of Q-Witt spaces (called there simply “Witt spaces”). The motivation for this definition is that such spaces possess intersection homology Poincaré duality IHi(X;K) ∼= Hom(IHn−i(X;K), K). The author’s paper [1] concerns K-Witt spaces and, in particular, a computation of the bordism theory ΩK−Witt ∗ of such spaces. However, there is an error in [1] 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 [11] for QWitt spaces by first performing a surgery to make the space irreducible and then performing ∗This work was partially supported by a grant from the Simons Foundation (#209127 to Greg Friedman) 2000 Mathematics Subject Classification: 55N33, 57Q20, 57N80