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Embeddings of non-simply-connected 4-manifolds in 7-space. II. On the smooth classification
- Source :
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics. 152:163-181
- Publication Year :
- 2021
- Publisher :
- Cambridge University Press (CUP), 2021.
-
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 indeterminancy. 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 $H_1\ne0$, with an indeterminancy. 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 $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z_{12}$.<br />Comment: 19 pages, exposition improved
- Subjects :
- Group (mathematics)
General Mathematics
010102 general mathematics
Geometric Topology (math.GT)
57R52, 57R67, 55R15
Mathematics::Geometric Topology
01 natural sciences
Connected sum
010101 applied mathematics
Combinatorics
Mathematics - Geometric Topology
Simply connected space
FOS: Mathematics
Torsion (algebra)
Isotopy
Algebraic Topology (math.AT)
Mathematics - Algebraic Topology
0101 mathematics
Invariant (mathematics)
Signature (topology)
Quotient
Mathematics
Subjects
Details
- ISSN :
- 14737124 and 03082105
- Volume :
- 152
- Database :
- OpenAIRE
- Journal :
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- Accession number :
- edsair.doi.dedup.....fef124637cc5cca84000ded498a88336