Which compact boundaryless 3-manifolds embed smoothly in the 4-sphere?

Importance: High ✭✭✭
Author(s): Kirby
Subject: Topology
Recomm. for undergrads: no
Posted by: rybu
on: November 7th, 2009

\begin{problem} Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smoothly in the 4-sphere. This should include a constructive procedure to find an embedding if the manifold is embeddable. % Enter your conjecture in LaTeX % You may change "conjecture" to "question" or "problem" if you prefer. \end{problem}

For general 3-manifolds this problem is fairly wide-open. But for some specific families of 3-manifolds it is heavily investigated.

There are two common embedding constructions: (1) obtain your 3-manifold as 0-surgery on a link which is the disjoint union of two smooth slice links. (2) Obtain your 3-manifold as the boundary of a Mazur manifold -- where Mazur manifold is taken to be a contractible 4-manifold constructed as $S^1 \times D^3$ union a 2-handle. In both cases the resulting 3-manifold M embeds smoothly in $S^4$. There are many other embedding constructions but no known "uniform" construction that works for all embeddable 3-manifolds.

Since such a 3-manifold would bound two 4-manifolds on either side, the embedding problem is a type of double cobordism problem, and related to issues such as the problem of determining which homology 3-spheres bound homology 4-balls.

The smoothness in the assumption is important. Mike Freedman has proven all homology 3-spheres admit tame topological embeddings into $S^4$. These embeddings have a less combinatorial nature than smooth embeddings so it is somewhat natural to restrict to the question of smooth embeddings. For example, the Poincare Homology Sphere does not embed smoothly in $S^4$, since it has a non-trivial Rochlin invariant.

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Bibliography

[B] R. Budney, \emph{Embeddings of 3-manifolds in the 4-sphere from the point of view of the $11$-tetrahedron census}, arXiv preprint arXiv:0810.2346

[CH] J.S. Crisp, J.A. Hillman, \emph{Embedding Seifert fibred $3$-manifolds and ${\rm Sol}\sp 3$-manifolds in $4$-space,} Proc. London Math Soc. (3) (1998), no. {\bf 3} 685--710.

[KK] A.~Kawauchi, S.~Kojima, \emph{Algebraic classification of linking pairings on $3$-manifolds,} Math. Ann. {\bf 253} (1980), no. 1, 29--42.

[FS] R.~Fintushel, R.~Stern, \emph{Rational homology cobordisms of spherical space forms,} Topology, {\bf 26} no. 3 pp. 385--393, (1987).

[GL] P.M.~Gilmer, C.~Livingston, \emph{On embedding 3-manifolds in 4-space,} Topology, {\bf 22}, no. 3, pp. 241--252 (1983).

*[K] Kirby, R. Problem list in low-dimensional topology. [http://math.berkeley.edu/~kirby/problems.ps.gz]

[L] R.A.~Litherland, \emph{Deforming twist-spun knots,} Trans. Amer. Math. Soc. {\bf 250} (1979), 311--331.

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)


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