Kirby Problem 4.107

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(Y,\xi)$ be a cooriented contact manifold of dimension $2n-1$: choose a one-form $\alpha$ with $\xi=\ker\alpha$ and $\alpha\wedge(d\alpha)^{n-1}$ nowhere zero. Its symplectization is \[ (\mathbb R\times Y,\omega),\qquad \omega=d(e^t\alpha). \] A hypersurface $H$ in a symplectic manifold $(M,\omega)$ is of contact type if there is a vector field $V$ near $H$ with $\mathcal L_V\omega=\omega$ that is transverse to $H$. The induced contact structure is $\ker((\iota_V\omega)|_{TH})$. A smooth isotopy of embeddings is a smooth family of embeddings with the given initial map. Homology below means singular homology with integral coefficients unless stated otherwise.

In dimension four, put $\omega_{\mathrm{std}}=dx_1\wedge dy_1+dx_2\wedge dy_2$ on $\mathbb R^4$ and let $\xi_{\mathrm{std}}$ on the unit $S^3$ be the kernel of the restriction of $\tfrac12\sum_{j=1}^2(x_jdy_j-y_jdx_j)$. A contact rational homology sphere is a closed contact three-manifold $Y$ with $H_*(Y;\mathbb Q)\cong H_*(S^3;\mathbb Q)$; identification of contact manifolds means a contactomorphism carrying their cooriented plane fields to each other.

Statement recorded in the supplied source.

Answer the three recorded questions:

1. (a) Characterize the embeddings $f:Y\hookrightarrow\mathbb R\times Y$ that induce isomorphisms $H_j(Y;\mathbb Z)\to H_j(\mathbb R\times Y;\mathbb Z)$ for every $j$ and can be smoothly isotoped to have contact-type image.
2. (b) Is every smooth embedded $S^3\subset(\mathbb R^4,\omega_{\mathrm{std}})$ smoothly isotopic to a contact-type hypersurface?
3. (c) Does a contact rational homology three-sphere not contactomorphic to $(S^3,\xi_{\mathrm{std}})$ embed in $(\mathbb R^4,\omega_{\mathrm{std}})$ as a contact-type hypersurface inducing its given contact structure?

Part (a) uses the contact manifold in the stated symplectization; parts (b) and (c) specifically use the standard four-dimensional ambient space. [S2]

Short English statement

When can a homology-preserving embedding in a symplectization be isotoped to contact type? In standard four-space, ask this for every embedded three-sphere and ask whether a different contact rational homology sphere can occur.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2983 (KP-4.107). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 R. İnanç Baykur, Robion C. Kirby and Daniel Ruberman, K3—A New Problem List in Low-Dimensional Topology, Mathematical Surveys and Monographs 295 (2026), author version, Problem 4.107 and Remarks, PDF page 279. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.