Embeddings of non-simply-connected 4-manifolds in 7-space. I. Classification modulo knots

Abstract

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a complete readily calculable classification of embeddings $N\to R^7$, up to the equivalence relation generated by isotopy and embedded connected sum with embeddings $S^4\to R^7$. Such a classification was already known only for $H_1=0$ by the work of Boéchat, Haefliger and Hudson from 1970. Our classification involves the Boéchat-Haefliger invariant $\varkappa (f)\in H_2$, Seifert bilinear form $\lambda (f):H_3\times H_3\to Z$ and $\beta$-invariant $\beta (f)$ which assumes values in a quotient of $H_1$ defined by values of $\varkappa (f)$ and $\lambda (f)$. In particular, for $N=S^1\times S^3$ we give a geometrically defined 1-1 correspondence between the set of equivalence classes of embeddings and an explicit quotient of the set $Z\oplus Z$. Our proof is based on development of Kreck modified surgery approach, involving some simpler reformulations, and also uses parametric connected sum.

52 pages, exposition improved

References in corpus (2)

• Embedding and knotting of manifolds in Euclidean spaces
• A classification of smooth embeddings of 3-manifolds in 6-space

Cited by in corpus (4)

• Classification of knotted tori
• On different reliability standards in current mathematical research
• A description of values of Seifert form for punctured n-manifolds in (2n-1)-space
• The classification of linked $3$-manifolds in $6$-space