 3-manifolds satisfying B3 but not B3'; see Section \ref{hypexsec} for additional remarks.


Returning to symplectic topology, Weimin Chen conjectured in \cite[Conjecture 5.1]{weimin} that the only rational homology sphere arising as the boundary of a rationally convex domain in $\mathbb{C}^2$ is the 3-sphere. In this direction, faced with the current lack of examples, we can ask:

\begin{question}\label{ctquestion} Does any rational homology 3-sphere, other than $S^3$, admit an embedding in $(\mathbb{R}^4, \omega_{std})$ as a contact type hypersurface?
\end{question}

Chen was motivated by questions in smooth 4-dimensional topology, particularly whether there exist smooth 4-manifolds homeomorphic but not diffeomorphic to the projective plane $\mathbb{C} P^2$. He shows that such an exotic 4-manifold would result if one were able to find a suitably ``small'' concave filling or ``cap'' of the circle bundle $M_{NS}$ over $\mathbb{R} P^2$ considered by Nemirovski-Siegel---essentially by attaching such a cap to the Stein domain bounded by $M_{NS}$. The proof that the result would not be diffeomorphic to $\mathbb{C} P^2$ relies on the fact that $M_{NS}$ cannot be realized as a hypersurface of contact type in $\mathbb{C} P^2$, which is a generalization of Nemirovski-Siegel's result that $M_{NS}$ does not bound a rationally convex domain in $\mathbb{C}^2$ (with nearly the same proof). While construction of small concave caps remains an open problem, and our techniques apply to domains in $\mathbb{C}^2$ rather than $\mathbb{C} P^2$, one may hope that the examples of Stein domains not having contact type boundary given here may allow some additional flexibility in this effort.

\begin{remark} As a partial response to Chen's conjecture concerning rationally convex domains with rational homology sphere boundary, we point out the following consequence of results in the literature:

\begin{theorem}\label{polyconvthm} If $W\subset \mathbb{C}^2$ is a polynomially convex domain whose boundary is a rational homology sphere, then $W$ is diffeomorphic to $B^4$ and in particular $\partial W \cong S^3$.
\end{theorem}

Here, the definition of polynomial convexity is identical to that of rational convexity with rational functions replaced by polynomials (with the same caveat regarding terminology as before). To prove the theorem, note that it is a consequence of work of Oka \cite{oka} (see also \cite[Remark 3.3]{CE-qconvex}) that if $W$ is as in the statement then one can add handles of index $\leq 2$ to $W$ in $\mathbb{C}^2$ to obtain a 4-ball $B^4\subset \mathbb{C}^2$. Then $B^4- \Int(W)$ is a rational homology cobordism from $\partial W$ to $S^3$, which is ribbon in the sense of \cite{DLVVW}. By \cite[Theorem 1.14]{DLVVW} (which is essentially a result of Gordon \cite{gordon81} combined with Perelman's geometrization theorem) it follows that $\partial W\cong S^3$, and in particular $W$ is a Stein filling of $S^3$. It was proved by Eliashberg \cite{E-filling} that any such filling is diffeomorphic to $B^4$. \hfill$\Box$
\end{remark}


Our obstruction to contact type embeddings can be described as follows. Recall that the Heegaard Floer homology of a compact, oriented 3-dimensional manifold $Y$ is a vector space over the field $\mathbb{F} = \mathbb{Z}/2\mathbb{Z}$, written $HF^+(Y)$. This vector space is always infinite-dimensional over $\mathbb{F}$, but has a natural finite-dimensional quotient $HF^+_{red}(Y)$, the {\it reduced} Heegaard Floer homology. If $\xi$ is a contact structure on $Y$, there is an associated element $c^+(\xi)\in HF^+(-Y)$ that is an invariant of $\xi$ up to contact isotopy, where $-Y$ denotes the 3-manifold $Y$ with the opposite orientation. If $\xi$ is overtwisted then $c^+(\xi) = 0$ while if $\xi$ is strongly fillable then $c^+(\xi)\neq 0$. Theorem \ref{obstrthm} below (particularly Corollary \ref{reducedcor}) shows that if $(Y,\xi)$ is embedded in $(\mathbb{R}^4, \omega_{std})$ as the boundary of a symplectically convex domain then, although $c^+(\xi)$ is nonzero, its image in $HF^+_{red}(-Y)$ vanishes. Our proof of Theorem \ref{mainthm} amounts to showing that, for any symplectically fillable contact structure on a Brieskorn sphere that bounds a 4-manifold with the integer homology of the 4-ball, this condition is not satisfied.


Unfortunately, the typical Brieskorn sphere admits {\it many} inequivalent fillable contact structures, and the obstruction just described applies to only one at a time. A key difficulty in the proof of Theorem \ref{mainthm} is that there is currently no general classification of contact structures on Brieskorn spheres. We take an approach that allows us to detect when the image of $c^+(\xi)$ in $HF^+_{red}(-Y)$ is nonzero, without particular assumptions on a symplectic filling of $\xi$. Our strategy is to study two numerical invariants associated to knots $K$ embedded in 3-manifolds: one, written $\tau_{sm}$, depends only on the (smooth) topology of $(Y,K)$, while the other, $\tau_\xi$, reflects aspects of the contact geometry. We observe that, from their definitions, if $c^+(\xi)$ vanishes in $HF^+_{red}(-Y)$, then these two invariants satisfy an inequality (Lemma \ref{taulemma}). Using a new estimate on the {\it twisting number} of any contact structure on a Brieskorn sphere (Theorem \ref{twistthm}), we show that this inequality is violated for any contact structure on a Brieskorn sphere that bounds a homology ball.






The paper is organized as follows.  In Section \ref{obstrsec1} we give the details of the obstruction to rational convexity mentioned above, and in Section \ref{obstrsec2} describe the invariants $\tau_{sm}$ and $\tau_\xi$. Section \ref{estimatesec} contains estimates on $\tau_{sm}$ and $\tau_\xi$ for the regular Seifert fiber in a Brieskorn sphere. In Section \ref{twistsec} we obtain a lower bound for the twisting number of any contact structure on a Brieskorn homology sphere, and this combined with results of Sections \ref{obstrsec} and \ref{estimatesec} proves Theorem \ref{mainthm}. In Section \ref{examplesec1} we deduce some additional consequences of our results for two questions that, it turns out, can be addressed by similar methods: namely, whether a contact structure is supported by an open book with pages of genus 0, and the existence a strong symplectic cobordism to the 3-sphere. The remainder of Section \ref{examplesec} provides some further details and examples regarding the relations between the conditions B1, B2, B3, B3' and B4, including illustrations of the use and limitations of our methods. The reader with particular interest in these questions, including examples of Stein domains in $\mathbb{C}^2$ that cannot be made Weinstein with respect to $\omega_{std}$, may read Sections \ref{examplesec2} through \ref{unobstrsec} first.

\subsection*{Acknowledgements} The authors are grateful to Bob Gompf for his interest in this project and many useful remarks and suggestions. Our thanks to Stefan Nemirovski for helpful comments on an early draft of this paper. We are also grateful to the referees for their careful reading and many suggestions. Part of this research was carried out in July 2019 while the authors were attending a PCMI workshop on ``Quantum Field Theory and Manifold Invariants". The authors are grateful to PCMI for the support and wonderful research environment. The first author was supported in part by a grant from the Simons Foundation (523795, TM).  Part of the research included in this article was carried out during the second author’s research stay in Montreal in Fall 2019. This research visit was supported in part by funding from the Simons Foundation and the Centre de Recherches Mathmatiques, through the Simons-CRM scholar-in-residence program. The second author is grateful to CRM and CIRGET, and in particular to Steve Boyer for their wonderful hospitality. The second author was also supported in part by grants from  NSF DMS-2105525 and the Simons Foundation (636841, BT).

\section{Floer invariants and contact type embeddings}\label{obstrsec}

\subsection{An obstruction to contact type embedding}\label{obstrsec1}  Recall \cite{OS:hf1,OS:hf2} that a closed