A homotopy equivalence that is not homotopic to a topological embedding
The authors show that for \(n\geq 6\), there is an open subset \(W^n\subset S^n\), such that it is homotopy equivalent to \(S^2\times S^{n-4}\), but there is no compact subset \(X^n\subset W^n\) such that the inclusion \(X^n \to W^n\) is a Čech equivalence. Similarly they obtain that there exists an open subset \(W^4\subset S^4\) such that it is homotopy equivalent to \(S^4\), but there is no compact subset \(X^4\subset W^4\) such that the inclusion \(X^4\to W^4\) is a \(\Check{\text{C}}\)ech equivalence. Using these two results, they construct interesting examples of open manifolds that have unusual compactness properties.
scientific article; zbMATH DE number 1307908
scientific article