# Questions about a paper by Laudenbach and Poénaru

I am working on the 1972 paper A Note on 4-Dimensional Handlebodies by F. Laudenbach and V. Poénaru, and I had two questions. I will use their notations to simplify things, since the paper is very short (8 pages long) and is easy to read.

First, I am willing to prove that any diffeomorphism $$\phi$$ of $$\#^k(S^1\times S^2)$$ can be extended to a diffeomorphism of $$\natural^k(S^1\times B^3)$$. By making use of their Lemma 1, and by inspecting the proof, I have set $$h=\beta^{-1}\phi^{-1}\alpha$$, to use what they did at page 342 of the journal (the article's sixth page). If we assume that $$\beta H_1 h\alpha^{-1}=\beta H_1\beta^{-1}\phi^{-1}$$ is orientation-preserving, then we are done, because of the remark they make : “mark that no diffeomorphism of $$X^p$$ was needed here !”.

However, is it always possible to assume that we are in this setting ? More precisely, if I require $$\phi$$ to be orientation-preserving from the start, it only suffices to have $$H_1$$ to be orientation-preserving. This $$H_1$$ is the restriction of $$H\in\text{Diff}(Y^p)$$, which was obtained from Lemma 2 by surjectivity. Is it always possible to assure that this $$H$$ is orientation-preserving ? Or even better : Is it always possible to prescribe the orientation of $$H$$ (to force it to be the same as $$\phi$$) ?

If I understand well their proof for that Lemma 2, they are obtaining the Nielsen transformations that generate $$\text{Aut}(\pi_1Y^p)$$. I have worked out that

• the permutation of two generators is obtained by permuting the corresponding handles in the decomposition of $$Y^p$$,
• and the one that inverses one generator is obtained by a diffeomorphism of the corresponding handle. Edit : now that I think about it, such a diffeomorphism would be orientation-reversing (wouldn't it ?), which is problematic regarding what follows !

For the third one, it seems to me that it boils down to a handle sliding. Is that correct to say that all these elementary diffeomorphisms are orientation-preserving, and thus that we can always choose $$H$$ to be this way ? And for the case where we need $$H$$ to be orientation-reversing, can we simply use an additional elementary diffeo that is orientation-reversing on $$Y^p$$ but that acts trivially on the free group ?

Edit : I will comment my own question with the remark that I find it unlikely to be able to prescribe the orientedness of $$H$$, for the reason that in their proof, Laudenbach and Poénaru didn't use this. If that were possible, they'd just pick $$H$$ so that their map is orientation-preserving and they are in the favorable case all the time... So what is wrong with my arguments regarding the Nielsen transformations ?

Now, my next question is probably easy, yet I cannot seem to see it... I have read several times that the paper implies that if a (closed, oriented) 4-manifold has a handle decomposition with one 0- and 4-handles, $$g$$ 1- and 3-handles, and no 2-handles, then it has to be diffeomorphic to $$\#^g(S^1\times S^3)$$. I totally agree with the case $$g=0$$, which is just copy-pasting their Theorem A'. However, I cannot see how this implies this more general version of the statement... How to go from the genus zero case to the more general case ?

A related question that may help me in understand what I'm missing is how to justify another I read : “the remarkable paper [...] tells you that there is only one way to attach the 3-handles”. Why is it so ? (it also seems it is a consequence of my first question, as this document relates in section 3.5 page 13... And yet I cannot see the link)

• In your penultimate paragraph, that closed manifold had best be diffeomorphic to a g-fold connected sum of $S^1 \times S^3$. Apr 21, 2021 at 18:31
• @DannyRuberman You're 100% correct, that's a typo on my side, it's fixed now, thanks ! Apr 21, 2021 at 18:41