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In 1956, Papakyriakopoulos proved Dehn's lemma, loop theorem and the sphere theorem. The proofs are based on a clever technique called "tower construction". Later, Whitehead, Shaprio, Stalling and Epstein adapted this idea to generalize the theorems, but still in the realm of 3-manifolds.

Are there any (relatively direct) generalizations in dimension at least 4?

Similar concepts to Dehn's lemma and the loop theorem have been explored. For instance, the h-cobordism theorem, especially in the context of simply connected manifolds, can be viewed as a generalization of ideas related to Dehn's lemma. One can also argue that the sphere theorem is related to surgery theory. But it seems all of them are far-reaching. Perhaps a clear-cut generalization is impossible. I would also appreciate if someone could shed some light on this.

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    $\begingroup$ There ought to be a (codimension 1) sphere theorem in dimensions at least 6 for manifolds with infinite fundamental group and trivial $\pi_k, 1 < k< n-1$ . Stallings’ theorem implies that the fundamental group of an n-manifold with nontrivial $\pi_{n-1}$ acts on a tree. Then one obtains an embedded $n-1$ submanifold whose image of the fundamental group is trivial (so it lifts to the universal cover). Then one ought to be able to surger away the fundamental group of the sub manifold and higher homotopy groups to make it geometrically simply connects and hence a sphere. $\endgroup$
    – Ian Agol
    Commented Nov 24, 2023 at 12:23

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I have a thread on the co-dimension $1$ generalization of Dehn's Lemma for 4-manifolds:

A problem/conjecture related to 4-manifolds that deserves a name. What name does it deserve?

My pair of papers with Gabai is about a simpler case where there is a type of failure. Not on existence of the spanning disc, but on the uniqueness. We show there are smooth embeddings $D^{n-1} \to S^1 \times D^{n-1}$ which agree with the standard inclusion $\{1\} \times S^{n-2} \subset S^1 \times D^{n-1}$ on the boundary, but which are not isotopic to the standard inclusion.

Our first paper is primarily about the $n=4$ case. The second paper argues these embeddings are topologically non-trivial. It also gives a detailed parametric argument valid for $n \geq 4$ -- but results are in $\pi_0$ of the embedding space only when $n=4$ in this paper. Our 3rd paper (not yet out) gives examples for all $n \geq 4$ in $\pi_0$, although the $n \geq 6$ examples were originally due to Hatcher and Wagoner (conjectured by Farrell). For $n \geq 6$ we show the Hatcher-Wagoner examples coincide with the Farrell examples, and the $n=5$ case is new, although I believe Kiyoshi Igusa might soon have a paper covering the $n=5$ case, extending the Hatcher-Wagoner techniques.

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  • $\begingroup$ Many thanks, Ryan. Are there any ways to relate your work to the codimension 1 sphere theorem? $\endgroup$
    – Shijie Gu
    Commented Dec 1, 2023 at 11:50
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    $\begingroup$ The group $\pi_0 Emb(D^{n-1}, S^1 \times D^{n-1})$ contains three pieces of information: a) the mapping class group of $D^{n-1}$, b) the generalized Schoenflies theorem/conjecture according to dimension and c) the information about whether or not such embeddings can be made disjoint from a linearly embedded copy, i.e. like $\{ -1 \} \times D^{n-1}$. None of these are exactly the sphere theorem, but (b) and (c) are certainly closely related, in that Alexander's Theorem in dimension 3 is certainly closely related to the sphere theorem. $\endgroup$
    – Ryan Budney
    Commented Dec 1, 2023 at 22:10
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Aru Ray and Danny Ruberman wrote a paper (here the arxiv version) about Dehn's lemma in dimension 4. From the abstract:

We investigate certain 4-dimensional analogues of the classical 3-dimensional Dehn's lemma, giving examples where such analogues do or do not hold, in the smooth and topological categories. In particular, we show that an essential 2-sphere $S$ in the boundary of a simply connected 4-manifold $W$ such that $S$ is null-homotopic in $W$ need not extend to an embedding of a ball in $W$. However, if $W$ has abelian fundamental group with boundary a homology sphere, then $S$ bounds a topologically embedded ball in $W$. Additionally, we give examples where such an $S$ does not bound any smoothly embedded ball in $W$. In a similar vein, we construct incompressible tori $T\subset \partial W$ where $W$ is a contractible 4-manifold such that $T$ extends to a map of a solid torus in $W$, but not to any embedding of a solid torus in $W$. Moreover, we construct an incompressible torus $T$ in the boundary of a contractible 4-manifold $W$ such that $T$ extends to a topological embedding of a solid torus in $W$ but no smooth embedding.

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