A classification of smooth embeddings of 4-manifolds in 7-space, II

Diarmuid Crowley; Arkadiy Skopenkov

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arXiv · 0808.1795

A classification of smooth embeddings of 4-manifolds in 7-space, II

Diarmuid Crowley·Arkadiy Skopenkov

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Abstract

Let N be a closed, connected, smooth 4-manifold with H_1(N;Z)=0. Our main result is the following classification of the set E^7(N) of smooth embeddings N->R^7 up to smooth isotopy. Haefliger proved that the set E^7(S^4) with the connected sum operation is a group isomorphic to Z_{12}. This group acts on E^7(N) by embedded connected sum. Boechat and Haefliger constructed an invariant BH:E^7(N)->H_2(N;Z) which is injective on the orbit space of this action; they also described im(BH). We determine the orbits of the action: for u in im(BH) the number of elements in BH^{-1}(u) is GCD(u/2,12) if u is divisible by 2, or is GCD(u,3) if u is not divisible by 2. The proof is based on a new approach using modified surgery as developed by Kreck.

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Diarmuid Crowley, Arkadiy Skopenkov. 2008-08-13. A classification of smooth embeddings of 4-manifolds in 7-space, II. https://arxiv.org/abs/0808.1795

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