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A. Skopenkov, Embedding and knotting of manifolds in Euclidean spaces, in: Surveys in Contemporary Mathematics, Ed. N. Young and Y. Choi, London Math. Soc. Lect. Notes, 347 (2008) 248-342. Available at the arXiv:0604045.

[edit] 1 Comment

The last sentence of subsection `The Generalized Haefliger-Wu invariant' of \S5 stated that for each subgroup G of S_p, the complex/manifold N and the number m from the Non-embeddability Examples 5.9 there is a G-equivariant map \widetilde N_G\to\widetilde{\R^m}_G. Only a weaker statement for \widetilde N_G replaced by its simplicial version is obvious. For Example 5.9.c the weaker statement follows from the existence of an almost embedding f:N\to\R^m [Segal&Skopenkov&Spiez1998, Example in p. 338]. Those G-equivariant maps are defined as f^p. Thus those maps are coherent for different G in the sense described in [Arone&Krushkal2021].

The stated assertion is presumably correct, but the proof could require particular properties of the almost embeddings constructed in the example.

In the last paragraph of this subsection it is stated that \alpha_G is not injective for each G (and some N,m). Analogously, only a weaker statement for \widetilde N_G replaced by its simplicial version is obvious.

[edit] 2 References

  • [Arone&Krushkal2021] G. Arone, V. Krushkal, Embedding obstructions in \R^d from the Goodwillie-Weiss calculus and Whitney disks, arxiv preprint.
  • [Segal&Skopenkov&Spiez1998] J. Segal, A. Skopenkov and S. Spie\. z, Embeddings of polyhedra in \R^m and the deleted product obstruction, Topol. Appl. 85 (1998), 225-234.
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