Cylindrical contact homology is a comparatively simple incarnation of symplectic field theory whose existence and invariance under suitable hypotheses was recently established by Hutchings and Nelson. We study this invariant for a general Brieskorn 3-manifold \(\Sigma(a_1,\ldots, a_n)\), and give a complete description of the cylindrical contact homology for this 3-manifold equipped with its natural contact structure, for any \(a_j\) satisfying \(\frac{1}{a_1} + \cdots + \frac{1}{a_n} < n-2\).

We consider constraints on the topology of closed 3-manifolds that can arise as hypersurfaces of contact type in standard symplectic \({\mathbb R}^4\). Using an obstruction derived from Heegaard Floer homology we prove that no Brieskorn homology sphere admits a contact type embedding in \({\mathbb R}^4\), a result that has bearing on conjectures of Gompf and Koll\'ar. This implies in particular that no rationally convex domain in \({\mathbb C}^2\) has boundary diffeomorphic to a Brieskorn sphere. We also give infinitely many examples of contact 3-manifolds that bound Stein domains but not symplectically convex ones; in particular we find Stein domains in \({\mathbb C}^2\) that cannot be made Weinstein with respect to the ambient symplectic structure while preserving the contact structure on their boundaries.

From a handle-theoretic perspective, the simplest  contractible 4-manifolds, other than the 4-ball, are Mazur manifolds. We produce the first pairs of Mazur manifolds that are homeomorphic but not diffeomorphic.  Our diffeomorphism obstruction comes from our proof that the knot Floer homology concordance invariant \(\nu\) is an invariant of the trace of a knot \(K \subset S^3\), i.e. the smooth 4-manifold obtained by attaching a 2-handle to \(B^4\) along \(K\). This provides a computable, integer-valued diffeomorphism invariant that is effective at distinguishing exotic smooth structures on knot traces and other simple 4-manifolds, including when other adjunction-type obstructions are ineffective.  We also show (modulo forthcoming work of Ozsváth and Szabó) that the concordance invariants \(\tau\) and \(\epsilon\) are not knot trace invariants. As a corollary to the existence of exotic Mazur manifolds, we produce integer homology 3-spheres admitting two distinct \(S^1\times S^2\) surgeries, resolving a question from Problem 1.16 in Kirby's list.

For an integer \(n\), write \(X_n(K)\) for the 4-manifold obtained by attaching a 2-handle to the 4-ball along the knot \(K\subset S^3\) with framing \(n\). It is known that if \(n< \overline{\mathrm{tb}}(K)\), then \(X_n(K)\) admits the structure of a Stein domain, and moreover the adjunction inequality implies there is an upper bound on the value of \(n\) such that \(X_n(K)\) is Stein. We provide examples of knots \(K\) and integers \(n\geq \overline{\mathrm{tb}}(K)\) for which \(X_n(K)\) is Stein, answering an open question in the field. In fact, our family of examples shows that the largest framing such that the manifold \(X_n(K)\) admits a Stein structure can be arbitrarily larger than \(\overline{\mathrm{tb}}(K)\). We also provide an upper bound on the Stein framings for \(K\) that is typically stronger than that coming from the adjunction inequality.