Abstract
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
${\mathbb Z} \oplus {\mathbb Z} \oplus {\mathbb Z} _{12}$
.
Publisher
Cambridge University Press (CUP)
Reference24 articles.
1. Local knotting of submanifolds (in Russian);Viro;Mat. Sbornik,1973
2. When is the set of embeddings finite up to isotopy?
3. Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots;Crowley;Moscow Math. J
4. A classification of smooth embeddings of 3-manifolds in 6-space
5. Classification problems in differential topology. V
Cited by
1 articles.
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