Some duality properties of non-saddle sets (Q5936525)

According to a theorem of \textit{B. Günther} and \textit{J. Segal} [Proc. Am. Math. Soc. 119, No. 1, 321-329 (1993; Zbl 0822.54014)]: A finite-dimensional compactum can be an attractor of a continuous flow on a manifold \(M\), if and only if it has the shape of a finite polyhedron. NEWLINENEWLINENEWLINEThe present authors prove the same assertion for an isolated non-saddle set instead of an attractor. Moreover they extend this result for locally compact ANRs instead of manifolds. Assume that the manifold \(M\) is an \(n\)-manifold, \(n>1\). Then: NEWLINENEWLINENEWLINELet \(K\) be an isolated non-saddle set of a continuous flow on \(M\). If \(K\) has trivial shape, then \(K\) is an attractor or repeller. NEWLINENEWLINENEWLINEUnder certain conditions the Conley index of an isolated non-saddle set \(K\) is uniquely determined by the shape of \(K\). The authors introduce the concept of a dual of a regular isolated non-saddle set \(K\) and give conditions under which the shape of \(K\) determines the shape of the dual. NEWLINENEWLINENEWLINEMore precisely: If \(K_i\) are regular isolated non-saddle sets of two continuous flows \(\varphi_i\), \(i=1,2\) on \(S^n\). If \(\text{Sh}(K_1)= \text{Sh}(K_2)\), then under certain conditions the same holds for the shape of the duals. NEWLINENEWLINENEWLINEThis result requires application of some shape complement theorems.