Questions in Geometric Group Theory — Q 9.2

Definitions and mathematical statement

An elementary collapse of a CW complex removes an open cell together with a free codimension-one face; its inverse is an elementary expansion. A simple homotopy equivalence between finite CW complexes is a homotopy equivalence obtained, up to homotopy, from finitely many such moves, with intermediate complexes of unrestricted dimension.

Let $K$ and $L$ be finite two-dimensional CW complexes which are simple homotopy equivalent. Can $K$ be transformed into $L$ by a finite sequence using only expansions and collapses of cell pairs of dimensions $(0,1)$ or $(1,2)$, together with homotopies of the attaching maps of two-cells?

For the last operation, retain the existing one-skeleton and replace an attaching loop \[ a:S^1\longrightarrow K^{(1)} \] by a loop homotopic to it in the subcomplex consisting of the one-skeleton and the other two-cells. This allows a two-cell to slide over another two-cell while its own attaching map is changed. All intermediate complexes remain at most two-dimensional; subdivision and cellular identifications are harmless changes of CW description.

The problem is the topological Andrews--Curtis question in the source. It does not permit an unrecorded three-cell expansion, since eliminating such higher-dimensional moves is exactly the issue. Nor is ordinary homotopy equivalence alone the hypothesis: the given simple homotopy equivalence excludes a Whitehead-torsion obstruction at the outset.

Short English statement

Can every simple homotopy equivalence between finite two-complexes be realized using only two-dimensional moves and two-cell slides?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100902 (AMR-010-0902), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 9.2 (PDF page 18). Original source indexed by AMR-010-0902. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.