Kirby Problem 4.45

Definitions and mathematical statement

Let $X$ be a closed simply connected smooth four-manifold and $\Sigma$ a closed oriented surface with a chosen smooth embedding $f:\Sigma\hookrightarrow X$. Remove an open ball disjoint from its image and call the resulting manifold $X^\circ$. Let $\mathcal E_{\mathrm{sm}}$ be the space of smooth embeddings $\Sigma\to\operatorname{int}(X^\circ)$ with the $C^\infty$ topology, and $\mathcal E_{\mathrm{top}}$ the space of locally flat embeddings with the compact-open topology. Use $f$ as basepoint.

Inclusion gives $i:\mathcal E_{\mathrm{sm}}\to\mathcal E_{\mathrm{top}}$. Attach punctured copies of $S^2\times S^2$ in a collar outside the original punctured manifold. This fixes inclusions into the successive stabilized ambient manifolds and defines stabilization maps on embedding spaces. For which $X,\Sigma,f$ and integers $k\ge0$ does \[ \ker i_* = \ker\left( \pi_k(\mathcal E_{\mathrm{sm}};f)\longrightarrow \operatorname*{colim}_{N}\pi_k\left( \operatorname{Emb}^{\mathrm{sm}}(\Sigma,X^\circ\#_N(S^2\times S^2));f\right)\right) \] hold? The left kernel uses the homomorphism to $\pi_k(\mathcal E_{\mathrm{top}};f)$.

For $k=0$, these are preimages of the component of $f$, rather than kernels of groups. For $k>0$, a class is a based continuous family of embeddings parametrized by $S^k$. Vanishing in the colimit means vanishing after some finite number of stabilizations. Thus the question compares becoming trivial through locally flat embeddings with becoming trivial through smooth embeddings after enlarging the ambient manifold. All embeddings are parametrized, as in the displayed spaces.

Short English statement

Do topologically trivial families of surface embeddings coincide with those made smoothly trivial by finite stabilization?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2921 (KP-4.45), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 4.45. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.