Even-Codimension Stratifications of Codimension-Two Embeddings

Definitions and mathematical statement

A piecewise-linear (PL) manifold is a manifold equipped with a PL structure. Here simply connected means path connected with trivial fundamental group. A homotopy equivalence $M\simeq N$ means that there are continuous maps $f:M\to N$ and $g:N\to M$ such that $g\circ f\simeq\operatorname{id}_M$ and $f\circ g\simeq\operatorname{id}_N$. The notation $D^2$ denotes a two-dimensional disk; the source does not specify its boundary convention. An embedding is a map that is a homeomorphism onto its image; whether a PL embedding is required here is not specified.

Source question. Let $M$ be a simply connected PL manifold with $M\simeq N$. The supplied statement asserts that there is consequently an embedding \[ i:M\hookrightarrow N\times D^2. \] Can this embedding be stratified using only strata of even codimension?

Unresolved qualifications. The asserted implication from homotopy equivalence to the existence of this embedding is retained as a source assertion, not established here. The record does not specify the nature or dimension of $N$, the dimension of $M$, compactness or boundary hypotheses, or further conditions needed for that assertion. It also does not specify which embedding is intended, whether the question concerns every such embedding or the existence of a suitable one, or whether the embedding must represent the given homotopy equivalence under projection to $N$.

A stratum is a manifold piece of a stratification, but the admissible stratifications and the compatibility required with $i$ are not defined in the record. For a stratum $S$ in a manifold $X$ of dimension $d$, its ambient codimension is $d-\dim S$; even codimension means that this integer belongs to $2{\mathbb{Z}}_{\geq 0}$. The record does not identify whether codimension is measured in $M$, in $N\times D^2$, or in another stratified space, nor whether the image must be a union of ambient strata. Thus the question above preserves the supplied target, but a fully specified quantified formulation requires these choices to be resolved.

Short English statement

The question asks whether the asserted embedding of a simply connected PL manifold $M$, homotopy equivalent to $N$, into $N\times D^2$ admits a stratification with only even-codimensional strata. The hypotheses guaranteeing the embedding, the meaning of a compatible stratification, and the reference space for codimension remain unspecified. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001949$, source problem code catalogue entry, title Even-Codimension Stratifications of Codimension-Two Embeddings, in the Oberwolfach Reports Open Problems collection.
• Supplied report citation: Stratified Spaces: Joining Analysis, Topology and Geometry (2012), Oberwolfach Reports, pp. 3217--3286, DOI 10.4171/owr/2011/56. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2011/56. The record assigns proposed year 2011 and names no individual proposer.
• Supplied literature assessment, checked on 2026-08-21: the problem is labeled open; the assessment says the report poses this embedding-stratification question and that the checked literature supplies related even-codimension theories but no answer to this specific construction. The only listed checked source is the original OWR report linked above. This is supplied dated curation, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.
• The registry assigns statement status corrected_verified and states that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. These are supplied verification claims; the report has not been independently consulted for this draft.