Topological Isolation of Singularities in Witt Bordism

Definitions and mathematical statement

The supplied question concerns a geometric construction in rational Witt bordism. Its precise category and orientation conventions are not stated. The following gives a conditional formulation under the usual convention of compact, oriented PL stratified pseudomanifolds without codimension-one strata; these conventions require confirmation from the source.

For such an $n$-dimensional stratified pseudomanifold $X$, the rational Witt condition requires that, for every singular stratum of odd codimension $2k+1$, with link $L$ of dimension $2k$, \[ I^{\bar m}H_k(L;{\mathbb{Q}})=0, \qquad \bar m(c)=\left\lfloor\frac{c-2}{2}\right\rfloor. \] Here $I^{\bar m}H_k$ denotes intersection homology with lower middle perversity. In the absence of codimension-one strata, the relevant integers satisfy $k\geq 1$. An orientation means an orientation of the regular stratum.

Two closed $n$-dimensional rational Witt spaces $X$ and $Y$ are rational Witt bordant if there is a compact, oriented $(n+1)$-dimensional stratified pseudomanifold $W$ with collared boundary and an orientation-preserving identification \[ \partial W\cong X\sqcup(-Y), \] where the interior satisfies the rational Witt condition and the stratification near the boundary is the product of the boundary stratification with a collar. The notation $-Y$ means $Y$ with reversed orientation. A space has at worst isolated singularities when its singular locus consists of isolated points; an empty singular locus is allowed.

Construction problem. For every dimension $n\geq 0$ and every closed $n$-dimensional rational Witt space $X$ in the intended bordism category, give a purely topological construction of a closed rational Witt space $Y$ of dimension $n$, having at worst isolated singularities, together with a rational Witt bordism $W$ from $X$ to $Y$.

The supplied dated assessment says that the existence of such a bordant representative is known algebraically; the requested target is a purely topological construction. The record does not specify what methods are permitted by that phrase. It also does not settle whether the intended category is PL or topological, whether orientation is required, or whether closedness and compactness are implicit. Consequently, the conventional formulation above is conditional and requires source-level clarification.

Short English statement

Construct, by purely topological means, a rational Witt bordism from each rational Witt space to one whose singularities are all isolated, allowing a nonsingular representative. The supplied assessment distinguishes this construction problem from the algebraically known existence statement. The intended category, bordism conventions, and meaning of "purely topological" need confirmation.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30001946$, exact source problem code catalogue entry; title: Topological Isolation of Singularities in Witt Bordism. The record belongs to Oberwolfach Reports Open Problems; no individual proposer is supplied, and the proposed year is recorded as 2011.
• Supplied report citation: Stratified Spaces: Joining Analysis, Topology and Geometry (2012), Oberwolfach Reports, pp. 3217--3286, DOI 10.4171/owr/2011/56. Source URL and supplied literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2011/56.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the bordism fact is known algebraically while the purely topological construction remains the question. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This verification is reported by the supplied record and has not been independently performed here.