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.


The cited (early) work by Cerf proves that, given a submanifold Y in a manifold X, the obvious map Diff(X)->Emb(Y,X) is a locally trivial fibration.

I guess that Budney and Gabai mean the following. By Palais, all embeddings D^3->S^4 are isotopic. Hence, for i=0, 1, the complement C_i of a small open tubular neighborhood U_i of Delta_i in S^4 is diffeomorphic with the compact 4-ball B^4. One has two disjoint embeddings phi_i, psi_i of B^3 in the boundary S^3 of B^4=C_i, one orientation-preserving, the other orientation-reversing, representing the two sides of Delta_i. It remains to extend the diffeomorphism between C_0 and C_1 through U_0 and U_1. This amounts to find a diffeomorphism f:B^4->B^4 such that f o phi_0=phi_1 and f o psi_0=psi_1. The papers by Palais and Cerf precisely give this. The connexity of Diff_+(S^2), and the unicity of the tubular neighborhood up to isotopy, serve to arrange that the extension goes well on a small neighborhood of the 2-sphere bounding Delta_0 and Delta_1.

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