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Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 22 January 2021, pp. 163-181
- Print publication:
- February 2022
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- Article
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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 
${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$.
