A counterexample to Wall's D(2) problem
Wall's finite D(2) problem: is a finite complex of dimension three that looks two-dimensional to homology and cohomology with all coefficients homotopy equivalent to a finite complex of dimension two? Chain complexes twisted by a character of the fundamental group; Fox derivatives; an obstruction for complexes of dimension two proved by a tower of infinite cyclic covers; the finite plus construction; an explicit group that gives a counterexample. The new arguments are proved in full; of six standard theorems of algebraic topology, five are proved in the programme and the homotopy invariance of twisted homology is a named hypothesis. The course is an exposition of OpenAI's construction (2026).
Four lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Wall's D(2) problem and twisted homology
- The tower obstruction
- The finite plus construction
- The counterexample
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Frame fields and primary obstructions · Characteristic classes
- Cell complexes and homotopy equivalences (Cell complexes, fundamental groups and covering spaces)
- Fundamental groups of graphs and two-complexes (Cell complexes, fundamental groups and covering spaces)
- Universal covers and cellular chains (Cell complexes, fundamental groups and covering spaces)
- CW complexes, the fundamental group, covering spaces and homology (core course Algebraic Topology); group rings and modules.