Let \(M\) be a compact \(3\)-manifold-with-boundary such that \(H_1(M; \mathbb{Z})\) contains a torsion element (i.e. element of finite order). Prove that \(M\) cannot be embedded as a submanifold of \(S^3\). (Hint: if it could, we could decompose \(S^3 = M \cup_\Sigma N\) as a union of two compact \(3\)-manifolds, glued along their common boundary surface \(\Sigma\).)
Solution
1. Suppose for contradiction that \(M\) embeds as a compact submanifold of \(S^3\).
Identify \(M\) with its image in \(S^3\). Because a compact manifold has the homotopy type of a finite CW complex, its homology groups are finitely generated.
2. Alexander duality implies that \(H^2(M;\ZZ)\) is free abelian.
The embedded manifold \(M\) is compact and locally contractible, so Alexander duality gives \[ \widetilde H^2(M;\ZZ) \cong \widetilde H_0(S^3\setminus M;\ZZ). \] The group \(\widetilde H_0\) of any space is free abelian, since it is free on the path components modulo one diagonal copy of \(\ZZ\). Hence \(\widetilde H^2(M;\ZZ)\) is free abelian. In degree \(2>0\), reduced and unreduced cohomology agree, so \(H^2(M;\ZZ)\) is free abelian.
3. If \(H_1(M;\ZZ)\) has nonzero torsion, then \(H^2(M;\ZZ)\) has nonzero torsion.
The universal coefficient theorem for cohomology gives a natural short exact sequence \[ 0 \longrightarrow \operatorname{Ext}(H_1(M;\ZZ),\ZZ) \longrightarrow H^2(M;\ZZ) \longrightarrow \operatorname{Hom}(H_2(M;\ZZ),\ZZ) \longrightarrow 0. \] By 1, the finitely generated abelian group \(H_1(M;\ZZ)\) decomposes as \[ H_1(M;\ZZ)\cong \ZZ^r\oplus T \] with \(T\) finite. If \(T\ne0\), then \[ \operatorname{Ext}(H_1(M;\ZZ),\ZZ) \cong \operatorname{Ext}(T,\ZZ) \cong T \] is nonzero torsion. The left arrow in the UCT sequence is injective, so this torsion subgroup embeds in \(H^2(M;\ZZ)\).
4. Therefore \(M\) cannot embed in \(S^3\).
The hypothesis says that \(H_1(M;\ZZ)\) has nonzero torsion. By 3, \(H^2(M;\ZZ)\) then has nonzero torsion, while 2 says that \(H^2(M;\ZZ)\) is free abelian. This contradiction disproves the assumed embedding from 1.