Nontrivial Witt Pairings in $\mathbb Z_2$-Witt Spaces

Definitions and mathematical statement

Interpret $\mathbb Z_2$ as the field ${\mathbb{F}}_2=\mathbb Z/2\mathbb Z$. In the usual boundaryless stratified pseudomanifold formulation, a $\mathbb Z$-orientation of $X$ is an integral orientation of its regular top-dimensional stratum. Let $\bar m(c)=\lfloor(c-2)/2\rfloor$ denote lower middle perversity. A stratified pseudomanifold without codimension-one strata is an ${\mathbb{F}}_2$-Witt space if, for every singular stratum of odd codimension $2r+1$, its link $L$, of dimension $2r$, satisfies \[ I^{\bar m}H_r(L;{\mathbb{F}}_2)=0. \] Here $I^{\bar m}H_*$ denotes intersection homology with the indicated perversity and coefficients. The Witt condition identifies lower and upper middle intersection homology and, for a compact boundaryless oriented space in the usual setting of intersection-homology duality, supplies a nonsingular middle-dimensional intersection pairing.

For $\dim X=4j+2$, write this symmetric bilinear pairing as \[ \lambda_X:V_X\times V_X\longrightarrow{\mathbb{F}}_2, \qquad V_X=I^{\bar m}H_{2j+1}(X;{\mathbb{F}}_2). \] Use the symmetric bilinear convention for $W({\mathbb{F}}_2)$: it is the Witt group of finite-dimensional nonsingular symmetric bilinear forms over ${\mathbb{F}}_2$, modulo metabolic forms. A form is metabolic if it has a totally isotropic subspace of half its dimension. Under this convention $W({\mathbb{F}}_2)\cong\mathbb Z/2\mathbb Z$, and its nonzero element is represented by the one-dimensional form $\langle1\rangle$, given by $(a,b)\mapsto ab$.

With these conventions, and interpreting the unspecified dimension parameter existentially, the supplied question is: \[ \begin{gathered} \text{Does there exist an integer }j\geq0\text{ and a compact, boundaryless,}\\ \mathbb Z\text{-oriented }{\mathbb{F}}_2\text{-Witt space }X\text{ of dimension }4j+2\\ \text{such that }[V_X,\lambda_X]=[\langle1\rangle]\neq0 \quad\text{in }W({\mathbb{F}}_2)? \end{gathered} \] The source record does not explicitly specify the quantification of $j$, the precise category of stratified spaces, or whether boundary is excluded. Thus the displayed formulation is conditional on the stated conventional interpretation. If spaces with boundary are intended, an absolute middle-dimensional pairing need not be nonsingular, and additional boundary conditions or a different pairing must be specified. The record also leaves the characteristic-two Witt-group convention implicit; the formulation above uses symmetric bilinear forms, not quadratic refinements.

Short English statement

The question asks whether some compact integrally oriented ${\mathbb{F}}_2$-Witt space of dimension $4j+2$ has a middle-dimensional intersection pairing with nonzero symmetric bilinear Witt class over ${\mathbb{F}}_2$. The draft uses a boundaryless interpretation and existentially quantifies $j\geq0$; these conventions require confirmation. The supplied record labels the problem open, but that status has not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001947$, source problem code catalogue entry, from the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2011.
• Supplied source citation: Stratified Spaces: Joining Analysis, Topology and Geometry (2012), Oberwolfach Reports, pp. 3217--3286, DOI 10.4171/owr/2011/56. Supplied source URL: https://doi.org/10.4171/owr/2011/56.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2011/56. Literature checked at: 2026-08-21. This is supplied dated curation, not an independently verified literature assessment. The supplied assessment says that the source leaves the existence of the specified space with nontrivial Witt pairing unsettled, and assigns status "open."
• The supplied statement status is corrected_verified. Its accompanying verification note reports that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This is a supplied verification claim; no independent source consultation is asserted here.