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Let $\Sigma$ be a smooth, $n$-dimensional homology sphere. In a paper that I am reading, the author states that there exists a smooth homotopy $n$-sphere $S$, such that the connected sum $\Sigma \# S$ can be embedded into $\mathbb{R}^{n+1}$.

I honestly have no idea how to approach this problem. The author says that it is not difficult to see, but all theorems that I know of (like Whitney embedding theorem) do not really look promising to me. Any help is greatly appreciated!

EDIT: The author did not specify in which dimensions this should work, but I am only interested in the case $n \geq 6$.

The_Rookie
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1 Answers1

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The stated result is true as long as $n \neq 3$. For $n=1$ or $2$, this follows easily from the classification of curves and surfaces. The case $n \geq 4$ is much harder. A proof is sketched in this answer by Moishe Kohan. Here I will repeat the argument with a few more details. Everything below takes place in the smooth category.

The main ingredients

Theorem A: Let $\Sigma$ be a closed manifold of dimension $n \geq 4$. If $\Sigma$ embeds in a homotopy sphere of dimension $n+1$, then $\Sigma$ embeds in $S^{n+1}$.

Proof: This is a consequence of the h-cobordism theorem. Suppose that $\Sigma$ embeds in a homotopy sphere $M$ of dimension $n+1$. Then $\Sigma$ also embeds in the connected sum $M \# (-M)$, since we can do the connected sum away from $\Sigma$. By Lemma 2.4 and Lemma 2.3 of [2], $M \# (-M)$ is h-cobordant to the standard sphere $S^{n+1}$. Since $M \#(-M)$ has dimension $n+1 \geq 5$, it follows from the h-cobordism theorem that $M \# (-M)$ is diffeomorphic to $S^{n+1}$. We conclude that $\Sigma$ embeds in $S^{n+1}$. $\square$

Theorem B (Kervaire [1, Theorem 3]): Let $\Sigma$ be a homology $n$-sphere. If $n =4$, then $\Sigma$ is the boundary of a contractible manifold. If $n \geq 5$, then there is a homotopy $n$-sphere $X$ such that $\Sigma \# X$ is the boundary of a contractible manifold.

Proof of the result

Proposition: Let $\Sigma$ be a homology $n$-sphere. If $n = 4$, then $\Sigma$ embeds in $S^5$. If $n \geq 5$, then there is a homotopy $n$-sphere $X$ such that $\Sigma \# X$ embeds in $S^{n+1}$.

Proof: We do the case $n \geq 5$, the case $n =4$ being similar. By Theorem B, there exists a homotopy $n$-sphere $X$ such that $\Sigma \# X$ is the boundary of a contractible manifold $V$. Let $DV$ be the double of $V$. Then $DV$ contains a copy of $V$, hence $\Sigma \# X$ embeds in $DV$. Moreover, $DV$ is simply-connected by the Van-Kampen theorem and has the homology of $S^{n+1}$ by Mayer-Vietoris. By the Whitehead theorem, it follows that $DV$ is a homotopy sphere. By Theorem A, we conclude that $\Sigma \# X$ embeds in $S^{n+1}$. $\square$

References

[1] Kervaire, M. Smooth homology spheres and their fundamental groups. Trans. Amer. Math. Soc.144(1969), 67–72.

[2] Kervaire, M. and Milnor, J. Groups of homotopy spheres. I. Ann. of Math. (2)77(1963), 504–537.

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