# Moscow Mathematical Journal

Volume 21, Issue 1, January–March 2021 pp. 43–98.

Embeddings of Non-Simply-Connected 4-Manifolds in 7-Space. I. Classification Modulo Knots

**Authors**:
D. Crowley (1) and A. Skopenkov (2)

**Author institution:**(1) Institute of Mathematics, University of Aberdeen,
United Kingdom, and

University of Melbourne, Australia

(2) Moscow Institute of Physics and Technology, 141700, Dolgoprudnyi, Russia, and

Independent University of Moscow, 119002, Moscow, Russia

**Summary: **

We work in the smooth category.
Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where
$H_q:=H_q(N; \mathbb{Z})$.
The main result is *a complete readily calculable classification of embeddings* $N\to\mathbb{R}^7$,
up to equivalence generated by isotopies and embedded connected sums with embeddings $S^4\to\mathbb{R}^7$.
Such a classification was earlier known only for $H_1=0$ by Boéchat–Haefliger–Hudson 1970.
Our classification involves the Boéchat–Haefliger invariant $\varkappa(f)\in H_2$,
Seifert bilinear form $\lambda(f)\colon H_3\times H_3\to\mathbb{Z}$ and $\beta$-invariant assuming values in the quotient of $H_1$ defined by values of $\varkappa(f)$ and $\lambda(f)$.
In particular, for $N=S^1\times S^3$ we define geometrically a 1–1 correspondence between the set of
equivalence classes of embeddings and an explicitly defined quotient of $\mathbb{Z}\oplus\mathbb{Z}$.

Our proof is based on development of Kreck modified surgery approach, involving some elementary reformulations, and also uses parametric connected sum.

2010 Math. Subj. Class. Primary: 57R40, 57R52; Secondary: 57R67, 57Q35, 55R15.

**Keywords:**Embedding, isotopy, 4-manifolds, surgery obstructions, spin structure.

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