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The exterior of a smooth proper disk has the homology of a circle

Statement

Assume AC. Let $D \subset B^{4}$ be a smooth proper disk and let $X_{D}$ be its compact exterior. Then $X_{D}$ is connected, $H_{1} (X_{D}; Z) ≅ Z$ generated by the positive meridian, and $H_{i} (X_{D}; Z) = 0$ for every $i \geq 2$ .

Statement references

The disk exterior and positive meridian

Charles Livingston and Swatee Naik, Introduction to Knot Concordance, §3.2, pp. 52–55

Axiom of Choice

Facts & Assumptions

Given: Full AC, a smooth proper disk $D \subset B^{4}$ , its chosen tubular neighborhood $N (D)$ , and its exterior $X_{D}$ .

[A1]

Full AC is assumed wherever the cover and orientation conventions use the choice axiom. The Axiom of Choice

[F1]

The tubular neighborhood is $N (D) ≅ D^{2} \times D^{2}$ and $N (D) \cap X_{D} ≃ D^{2} \times S^{1} ≃ S^{1}$ , whose generator is the positive meridian. Exterior of a smooth proper disk

[F2]

Mayer–Vietoris is exact for the collar-thickened decomposition $B^{4} = N (D) \cup X_{D}$ . Mayer–Vietoris sequence in singular homology

Proof

1.1F1F2givenA1

Both $B^{4}$ and $N (D)$ are contractible, and $N (D) \cap X_{D}$ is connected and homotopy equivalent to $S^{1}$ by [F1]. The reduced degree-zero part of Mayer–Vietoris [F2] therefore gives ${\tilde{H}}_{0} (X_{D}; Z) = 0$ , so $X_{D}$ is connected.

2.1F1F2step 1.1algebra

In degree one, the sequence has $H_{2} (B^{4}) = 0$ , then $H_{1} (N (D) \cap X_{D}) ≅ Z$ , then $H_{1} (N (D)) \oplus H_{1} (X_{D})$ , then $H_{1} (B^{4}) = 0$ . Since $H_{1} (N (D)) = 0$ , the inclusion of the overlap induces an isomorphism $H_{1} (N (D) \cap X_{D}) \to H_{1} (X_{D})$ . Its generator is the positive meridian by [F1].

3.1F1F2step 2.1algebra∎

For every $i \geq 2$ , the adjacent groups $H_{i} (N (D) \cap X_{D})$ , $H_{i} (N (D))$ , and $H_{i} (B^{4})$ vanish. Exactness of [F2] then forces $H_{i} (X_{D}) = 0$ . This proves the asserted homology in all degrees.

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Sources

Charles Livingston and Swatee Naik, Introduction to Knot Concordance (standard reference, not scraped)