TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Topological realization of the E8 intersection form

Statement

Assume full AC. There exists a closed connected simply connected oriented topological four-manifold $Y_{E_{8}}$ whose integral intersection form is isometric to the positive-definite even unimodular lattice $L_{E_{8}}$ . This is a topological realization statement and does not assert a smooth structure on $Y_{E_{8}}$ .

Statement references

Freedman, The topology of four-dimensional manifolds, Theorems 1.5 and 1.7

Facts & Assumptions

Given: Full AC and the E8 lattice as defined by its Gram matrix. The run-specific TOP external-source permission in CLAUDE.md §12 applies to the following explicitly imported realization input.

[F1]

$L_{E_{8}}$ is positive definite, even, and unimodular. The E8 lattice is even unimodular and is not diagonal

[F2]

External TOP input — Freedman's realization theorem. For every integral unimodular symmetric form $ω$ , there exists an oriented closed simply connected almost-smooth topological four-manifold whose integral intersection form is $ω$ . Here almost-smooth means smooth off a single point. This is the existence clause of Freedman, The topology of four-dimensional manifolds, Theorem 1.5, printed p. 368; Theorem 1.7, printed p. 371, separately records the E8 case. This input is imported from that source under the run-specific TOP exception; its plumbing, contractible cap and four-dimensional realization proof are not reconstructed here.

Proof

Given: The local lattice calculation [F1] and the explicit external TOP realization input [F2].

1.1F1F2

Apply [F2] to the integral unimodular form $ω = L_{E_{8}}$ , which satisfies its exact algebraic hypothesis by [F1]. It gives an oriented closed simply connected topological four-manifold $Y_{E_{8}}$ with intersection form $L_{E_{8}}$ . Simply connected includes path connectedness, so $Y_{E_{8}}$ has every property required in the statement.

2.1F2step 1.1∎

The source also gives smoothness away from a point, but no smooth structure on the whole closed manifold is asserted or used. This establishes precisely the stated topological realization.

Imported-source boundary

The realization existence in [F2] is an external input permitted for this run's TOP branch, not a locally reconstructed proof or new acceptance of an arbitrary disk-embedding theorem. The lattice calculation remains the existing local supplier. No smooth diagonalization result is used to obtain the topological manifold.

Depends on

Used by

The topological E8 manifold does not satisfy the smooth diagonalization conclusion Counterexample

Dependency tree · two levels

6 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)