# 3-balls with the same boundary in $S^4$ differ up to diffeomorphism

I am looking at this recent paper by Budney and Gabai and I am confused by a certain sentence in it. Theorem 4.7 states that if $$\Delta_1$$ and $$\Delta_2$$ are two 3-balls smoothly embedded in $$S^4$$ that are identical near there boundary, then there is a diffeomorphism $$\Phi: S^4 \to S^4$$ that is the identity on the neighborhood of the boundary of these two balls where they agree and $$\Phi(\Delta_1) = \Delta_2$$.

The proof of this theorem is one sentence long, and this is the sentence where I am seeking some clarification. The sentence cites two facts:

(1) Regular neighborhoods are unique

(2) $$\operatorname{Diff}_0(S^2)$$ is connected

and two papers:

(1) J. Cerf, Topologie de certains espaces de plongements, Bull. Soc. Math. France, 89 (1961), 227–380.

(2) R. Palais, Extending diffeomorphisms, Proc. Amer. Math. Soc. 11 (1960), 274–277.

I am not sure exactly what parts of these two references are being used. I believe for the Palais paper, maybe the authors are using Corollary 2 to say that there is some diffeomorphism of $$S^4$$ taking $$\Delta_1$$ to $$\Delta_2$$. I am not sure what is in the Cerf paper (it is written in french and over 100 pages so that has kept me away). However, maybe this is the standard reference for the fact that every diffeomorphism of $$S^3$$ extends over $$B^4$$?

I'm also not exactly sure what the statement "regular neighborhoods are unique" means. I suppose it means that any two regular neighborhoods for a smooth submanifold differ up to homotopy rel the submanifold.

I would love it if someone could tell me how to fit these pieces together and understand the proof.

• I think you have your question answered. The issue your question perhaps did not emphasize is that while $\Delta_1$ and $\Delta_2$ are isotopic, they are generally not isotopic through embeddings that preserve the boundary, or slightly weaker, isotopic through embeddings that are linear inclusions on the boundary. May 5 at 21:46