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The topological E8 manifold does not satisfy the smooth diagonalization conclusion
Statement refuted
Under full AC, every closed connected simply connected oriented topological four-manifold with positive-definite integral intersection form has that form isometric over the integers to the standard diagonal lattice.
Statement references
Facts & Assumptions
Given: Full AC and the topological-category assertion in the Statement refuted.
There is a closed connected simply connected oriented topological four-manifold with intersection form . The supplier explicitly applies Freedman's external realization theorem under the run-specific TOP exception; no local reconstruction of that existence theorem is claimed. Topological realization of the E8 intersection form
The form is positive definite and unimodular but not isometric over the integers to the standard diagonal rank-eight lattice. The E8 lattice is even unimodular and is not diagonal
Counterexample
Given: Full AC, the asserted topological extension and the realization/lattice facts [F1]–[F2].
Take the manifold furnished by [F1]. It is closed, connected, simply connected, oriented and topological, so it meets the asserted manifold hypotheses.
Its intersection form is positive definite by [F2], but the same local lattice result proves it is not isometric to the standard diagonal form. Thus satisfies every hypothesis of the refuted assertion and violates its conclusion.
Sources
Remarks
Donaldson's diagonalization theorem assumes a smooth four-manifold. This topological realization does not meet that hypothesis, and the lattice calculation alone does not give a knot nonsliceness certificate.
Remarks references
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Michael H. Freedman, The topology of four-dimensional manifolds (standard reference, not scraped)