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The four-dimensional Singer conjecture

Poincaré duality reduces the four-dimensional Singer target to a vanishing assertion but does not establish that vanishing.

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Understanding the problem

Read the background

Poincaré duality reduces the four-dimensional Singer target to a vanishing assertion but does not establish that vanishing. Results for special aspherical classes leave the general target separate.

Dimension four and asphericity are required. This entry does not claim the Singer conjecture in every dimension or for nonaspherical manifolds.

Why it matters

If correct, establishes the stated middle-degree concentration throughout the four-dimensional aspherical class.

What counts as progress

A sourced resolution or substantial advance on the precise target, with hypotheses, prior context, and verification evidence stated separately.

What to watch for

  • Assess the precise statement and assumptions before comparing results.
  • Dimension four and asphericity are required. This entry does not claim the Singer conjecture in every dimension or for nonaspherical manifolds.
  • Track independent assessments, dependencies, corrections, and withdrawals separately from importance.

Background review: 2026-10-11. Selected sources:

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