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**A note on strong Jordan separation.**
*(English)*
Zbl 1171.57023

In a previous paper [J. Lond. Math. Soc., II. Ser. 73, No. 3, 681-700 (2006; Zbl 1167.53039)], the author established a result which he termed strong Jordan separation. This was a version of Jordan separation which applied to maps \(f:S^n\rightarrow S^{n+1}\) which are not assumed to be injective. Under some mild hypothesis, one could nevertheless ensure that the image separated \(S^{n+1}\), and that any continuous extension \(F:\mathbb{D}^{n+1}\rightarrow S^{n+1}\) surjects onto one of the connected components of \(S^{n+1}-f(S^n)\). In the present note the author extends his result to the broadest possible setting, by establishing the following two results:

Theorem 1. Let \(X\) be a compact topological space, \(f:X\rightarrow S^{n+1}\) a continuous map, and \(U\subset X\) an open subset homeomorphic to an open \(n\)-disk \(\mathbb{D}^n_{\circ}\). Assume that 4mm

(In Theorem 1, one should think of the sets \(V_1,V_2\) as corresponding locally to the two “sides” of \(f(U)\cong \mathbb{D}^n_{\circ}\) in the ambient \(S^{n+1}\).)

Theorem 2. Under the hypotheses of the previous theorem, let us further assume that \(X\) is a closed subspace of an ambient topological space \(\widehat{X}\). Define two subgroups of \(H_n(X;\mathbb{Z}_2)\) by: 4mm

The author also discusses some specific examples covered by the two theorems.

Theorem 1. Let \(X\) be a compact topological space, \(f:X\rightarrow S^{n+1}\) a continuous map, and \(U\subset X\) an open subset homeomorphic to an open \(n\)-disk \(\mathbb{D}^n_{\circ}\). Assume that 4mm

- \(\bullet\)
- the map \(f:X\rightarrow S^{n+1}\) contains \(U\) in its set of injectivity (i.e. \(U\subset \text{Inj}(f):=\{x\in X\mid f^{-1}(x)=x\}\)), and
- \(\bullet\)
- the map \(\check{H}^n(X;\mathbb{Z}_2)\rightarrow \check{H}^n(X-U;\mathbb{Z}_2)\) on Čech cohomology groups induced by the inclusion \(X-U\hookrightarrow X\) has a non-trivial kernel.

(In Theorem 1, one should think of the sets \(V_1,V_2\) as corresponding locally to the two “sides” of \(f(U)\cong \mathbb{D}^n_{\circ}\) in the ambient \(S^{n+1}\).)

Theorem 2. Under the hypotheses of the previous theorem, let us further assume that \(X\) is a closed subspace of an ambient topological space \(\widehat{X}\). Define two subgroups of \(H_n(X;\mathbb{Z}_2)\) by: 4mm

- \(\bullet\)
- \(K=\ker(H_n(X;\mathbb{Z}_2)\rightarrow H_n(\widehat{X};\mathbb{Z}_2))\), and
- \(\bullet\)
- \(J=\text{im}(H_n(X-U;\mathbb{Z}_2)\rightarrow H_n(X;\mathbb{Z}_2))\),

The author also discusses some specific examples covered by the two theorems.

Reviewer: Ioan Pop (Iaşi)

### MSC:

57N99 | Topological manifolds |

55M05 | Duality in algebraic topology |

57N45 | Flatness and tameness of topological manifolds |

### Citations:

Zbl 1167.53039
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\textit{J.-F. Lafont}, Publ. Mat., Barc. 53, No. 2, 515--525 (2009; Zbl 1171.57023)

### References:

[1] | F. D. Ancel, Resolving wild embeddings of codimension\guioone manifolds in manifolds of dimensions greater than \(3\), Special volume in honor of R. H. Bing (1914\Ndash1986), Topology Appl. 24(1-3) (1986), 13\Ndash40. · Zbl 0607.57011 |

[2] | F. D. Ancel and J. W. Cannon, The locally flat approximation of cell-like embedding relations, Ann. of Math. (2) 109(1) (1979), 61\Ndash86. · Zbl 0405.57007 |

[3] | R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 465\Ndash483. · Zbl 0079.38805 |

[4] | T. Iwaniec and J. Onninen, Deformations of finite conformal energy: existence, and removability of singularities, Proc. London Math. Soc. (to appear). · Zbl 1196.30023 |

[5] | J.-F. Lafont, Strong Jordan separation and applications to rigidity, J. London Math. Soc. (2) 73(3) (2006), 681\Ndash700. · Zbl 1167.53039 |

[6] | S. Lefschetz and J. H. C. Whitehead, On analytical complexes, Trans. Amer. Math. Soc. 35(2) (1933), 510\Ndash517. · Zbl 0006.37006 |

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