TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-10-22 (gpt-6.1-sol)

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Standardized Casson towers yield locally flat disks

Statement

Let $X$ be a compact oriented topological $4$ -manifold with nonempty boundary. A kinky handle is a standard $2$ -handle whose core disk has finitely many transverse self-plumbings; a framed Casson-handle tower is a nested sequence of such stages in which each self-plumbing point of one stage is replaced by a framed kinky handle at the next stage, with the attaching regions matching the preceding stage. Suppose a finite family of these tower $4$ -neighborhoods $T_{i}$ , with specified attaching regions $A_{i}$ , is embedded pairwise disjointly as relatively open subsets of $X$ , with $T_{i} \cap \partial X = A_{i}$ , and carries the algebraic dual data of a disk-embedding datum. If each embedded tower has a homeomorphism of pairs to the standard open $2$ -handle $D^{2} \times R^{2}$ that takes its specified framed attaching region, including its core circle and normal-framing class, to the standard attaching region $S^{1} \times R^{2}$ with its standard framing, then the inverse images of the cores $D^{2} \times {0}$ form a pairwise disjoint family of locally flat properly embedded disks in $X$ , with the prescribed framed boundary circles.

Statement references

The relative disk and dual-sphere interface

Stefan Behrens et al., The Disc Embedding Theorem, Chapters 17–18 and 24–28, https://doi.org/10.1093/oso/9780198841319.001.0001

Facts & Assumptions

Given: The stated family of towers and their relative standardization homeomorphisms.

[F1]

Freedman disk-embedding data: A disk-embedding datum specifies proper disk boundary circles and their boundary normal-framing classes.

Proof

1.1givenconstruct

A standard open $2$ -handle is $D^{2} \times R^{2}$ , and its core $D^{2} \times {0}$ is a locally flat properly embedded disk whose boundary is the core attaching circle with the standard normal-framing class.

2.1givenstep 1.1construct

Write $h_{i} : (T_{i}, A_{i}) \to (D^{2} \times R^{2}, S^{1} \times R^{2})$ for the supplied homeomorphism, and set ${\overset{ˉ}{f}}_{i} (z) = h_{i}^{- 1} (z, 0)$ . Since $T_{i}$ is relatively open in $X$ and $T_{i} \cap \partial X = A_{i}$ , the inverse of $h_{i}$ , followed by inclusion in $X$ , identifies the standard handle with a relatively open ambient neighborhood, including its boundary. Restricting its product charts along the core gives charts of $(X, {\overset{ˉ}{f}}_{i})$ modeled on $(R^{4}, R^{2})$ at interior points and $(R_{+}^{4}, R_{+}^{2})$ at boundary points. Moreover ${\overset{ˉ}{f}}_{i}^{- 1} (\partial X) = S^{1}$ , because $h_{i}^{- 1} (S^{1} \times R^{2}) = A_{i}$ . Thus these are locally flat proper disks in the ambient pair, not merely intrinsically in the towers.

3.1F1step 1.1step 2.1construct∎

The disks lie in disjoint $T_{i}$ , so are pairwise disjoint. The framing-preserving condition on $h_{i}$ transports the standard boundary frame to the prescribed boundary normal-framing class. The disks therefore have all the required boundary data.

Depends on

Used by

Freedman disk-embedding theorem for good groups Theorem

Dependency tree · two levels

3 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 and Frank Quinn, Topology of 4-Manifolds (standard reference, not scraped)
Stefan Behrens et al., The Disc Embedding Theorem (standard reference, not scraped)