We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; {\mathbb Z} )$. Our main result is a readily calculable classification of embeddings$N \to {\mathbb R}^7$up to isotopy, with an indeterminacy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9.The group of knots $S^4\to {\mathbb R}^7$ acts on the set of embeddings $N\to {\mathbb R}^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1 \ne 0$, with an indeterminacy.Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008.For $N=S^1\times S^3$ we give a geometrically defined 1–1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set ${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$.

中文翻译：

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在 7 空间中嵌入非简单连接的 4 流形。二、关于平滑分类

我们在平滑类别中工作。让$N$是一个无扭转的封闭连接的可定向 4 歧管$H_1$， 在哪里$H_q := H_q(N; {\mathbb Z} )$. 我们的主要结果是易于计算的嵌入分类$N \to {\mathbb R}^7$直到同位素，具有不确定性。这种分类以前只知道$H_1=0$通过我们从 2008 年开始的早期工作。我们的分类完成时$H_2=0$或者当签名$N$既不能被 64 也不能被 9 整除。结组$S^4\to {\mathbb R}^7$作用于嵌入集$N\to {\mathbb R}^7$通过嵌入的连接总和达到同位素。在第一部分中，我们对这个动作的商进行了分类。本文的主要新颖之处在于对这一行动的描述$H_1 \ne 0$，具有不确定性。除了第一部分的不变量外，检测结的动作还涉及对我们 2008 年工作的 Kreck 不变量的改进。对于$N=S^1\次 S^3$我们给出了嵌入的同位素类集合与该集合的某个明确定义的商之间的几何定义的 1-1 对应关系${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$.