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

Lafont, Jean-François

Publ. Mat., Barc. 53, No. 2, 515-525 (2009).

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

\(\bullet\)

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.

Then \(f(X)\) separates \(S^{n+1}\) into at least two connected components. Furthermore, there are precisely two connected components \(V_1,V_2\) of \(S^{n+1}-f(X)\) having the property that their closure \(\overline{V}_i\) intersects \(f(U)\). In fact, for these two connected components, we have containments \(f(U)\subset \overline{V}_i\).
(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

\(\bullet\)

\(J=\text{im}(H_n(X-U;\mathbb{Z}_2)\rightarrow H_n(X;\mathbb{Z}_2))\),

where both maps are induced by the corresponding inclusions of spaces. If \(K\nsubseteq J\), then we have that for any continuous extension \(F:\widehat{X}\rightarrow S^{n+1}\), \(F\) surjects onto one of the two components \(V_i\).
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

Keywords:

Jordan separation; Alexander duality; codimension one taming

Citations:

Zbl 1167.53039

References:

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