Journal article

EMBEDDINGS OF NON-SIMPLY-CONNECTED 4-MANIFOLDS IN 7-SPACE. I. CLASSIFICATION MODULO KNOTS

D Crowley, A Skopenkov

MOSCOW MATHEMATICAL JOURNAL | INDEPENDENT UNIV MOSCOW-IUM | Published : 2021

Abstract

We work in the smooth category. Let N be a closed connected orientable 4-manifold with torsion free H1, where Hq:=Hq(N; Z). The main result is a complete readily calculable classification of embed-dings N ? R7, up to equivalence generated by isotopies and embedded connected sums with embeddings S4 ? R7 . Such a classification was earlier known only for H1 = 0 by Boéchat–Haefliger–Hudson 1970. Our classification involves the Boéchat–Haefliger invariant ?(f) ? H2, Seifert bilinear form ?(f): H3 × H3 ? Z and ß-invariant assuming values in the quotient of H1 defined by values of ?(f) and ?(f). In particular, for N = S1 × S3 we define geometrically a 1–1 correspondence between the set of equivale..

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University of Melbourne Researchers

Grants

Awarded by Russian Foundation for Basic Research


Funding Acknowledgements

We would like to thank the Hausdorff Institute for Mathematics and the University of Bonn for their hospitality and support during the early stages of this project. A. Skopenkov is supported in part by the Russian Foundation for Basic Research Grant No. 15-01-06302, by Simons-IUM Fellowship and by the D. Zimin Dynasty Foundation.