# smooth homotopy 4-balls with sphere boundary in dimension 4

What follows is, as far as I can tell, totally standard folklore. I have one particular point of confusion, other than that, I wanted to confirm that I am uttering the incantations correctly.

The smooth 4-dimensional Poincare conjecture (SPC4) is the statement that any smooth 4-dimensional manifold $$\Sigma$$ that is homeomorphic to $$S^4$$ is diffeomorphic to $$S^4$$. Is SPC4 equivalent to the conjecture that any smooth 4-manifold $$B$$ with $$\partial B = S^3$$ that is homotopy equivalent to $$B^4$$ is diffeomorphic to $$B^4$$?

Going one direction: Assume SPC4 is true.

If $$B$$ is a homotopy 4-sphere with $$\partial B = S^3$$ then we can fill the boundary with $$B^4$$ and we obtain a simply connected (by van Kampen) homology $$S^4$$ (by Mayer-Vietoris), which (by Hurewicz and Whitehead) must be a homotopy $$S^4$$, which (by Freedman) must be homeomorphic to $$S^4$$, which (assuming SPC4) is then diffeomorphic to $$S^4$$.

I imagine that we are about ready to conclude that $$B$$ must be standard - since it is now sitting inside of $$S^4$$ with its boundary bounding a standard $$B^4$$ on the other side. How do we finish up? Can we ambiently isotope $$S^4$$ so as to have the image of the $$S^3$$ be just the usual $$S^3 \subset S^4$$?

Going the other direction: Assume that every smooth homotopy $$B^4$$ with boundary $$S^3$$ is diffeomorphic to $$B^4$$.

Suppose that $$\Sigma$$ is a smooth homotopy $$S^4$$. Take a small 4-ball $$B^4$$ in $$\Sigma$$ and remove it. What is left (by van Kampen) is a smooth homotopy $$B^4$$ with boundary $$S^3$$ which (by assumption) is then diffeomorphic to $$B^4$$. Now (by Cerf) since $$\Sigma$$ is just the union of two copies of $$B^4$$, it is in fact diffeomorphic to $$S^4$$.

Yes, you can perform that ambient isotopy: any oriented embedding $$i: B^n \to M^n$$ is isotopic to any other. (This is a lemma proven independently by Cerf and Palais1, but the idea is quite clear: shrink the image of $$i$$ until it's contained in the chart, then take the limit that defines the derivative of a map.)

In particular, if $$h$$ is your diffeomorphism $$B \cup_{\partial B} B^4 \to S^4$$, you may isotope the embedding of $$h(B^4)$$ so that it is the standard inclusion of the north hemisphere. Then $$h$$ restricts to a diffeomorphism $$h: B \to B^4$$, where this $$B^4$$ is the southern hemisphere of $$S^4$$.

1The references given on this Manifold Atlas page are Palais, Theorem 5.5 (the essential content is Lemma 5.2) and somewhere in Cerf's treatise on embedding spaces.

• Are you sure the fact about balls is is due Smale? This is usually called the Palais (196)-Cerf (1961) lemma. Nov 6, 2018 at 11:48
• @DannyRuberman No, but I'm unsure what I was thinking of! I will edit the reference.
– mme
Nov 6, 2018 at 14:00
• The specific reference in the Cerf paper is section 5.1.4. (It's easier to find if you search for "boule" rather than "disque" (or "bisque" as auto-correct would have it).) I think of the Palais result as being from Proc. Amer. Math. Soc. 11 1960 274–277, but you are right about the earlier one. Nov 6, 2018 at 16:25
• @DannyRuberman Thanks for your care in making sure the references were right.
– mme
Nov 6, 2018 at 16:37