Corrigendum: Intersection homology with field coefficients: K‐Witt spaces and K‐Witt Bordism

Abstract

We construct geometric examples of pseudomanifolds that satisfy the Witt condition for intersection homology Poincaré duality with respect to certain fields but not others. We also compute the bordism theory of K-Witt spaces for an arbitrary field K, extending results of Siegel for K = Q.

K-Witt bordism in characteristic 2

“…To see that the bordism is via a Witt space, it is only necessary to observe that the link of the interior cone point in each such pinch bordism will be a wedge of S 2 s, and it is easy to compute that ImH 1 (∨ i S 2 ; K) = 0 for any K. But now, since all closed oriented 4 surfaces bound, Ω Z 2 −Witt 2 = 0. This special case was also over-looked in [1], though this argument holds for any field K and is consistent with the claim of [1] that Ω K−Witt 2 = 0 for all K. Thus we have shown that w : [1,Corollary 4.3] that the bordism groups depend only on the characteristic of the field, so for characteristic 2 it suffices to consider K = Z 2 .…”

“…Putting this together with the computations from [1] of Ω K−Witt * in dimension ≡ 4k + 2 mod 4 (which remain correct), we have the following theorem:…”

“…As the intersection product is skew symmetric on ImH 2k+1 (X; K), such a [z] always exists. The error in [1] stems from overlooking that this last fact is not necessarily true in characteristic 2, where skew symmetric forms and symmetric forms are the same thing and so skew-symmetry does not imply [z] · [z] = 0.…”

“…Corrected computations. To begin to remedy the error of [1], we first observe that it remains true in characteristic 2 that the map 2 w :…”

“…Since writing [1], the author has discovered that this theorem is also provided without detailed proof by Goresky in [4, page 498]. We provide here the details:…”