Taut sets in three space are very special (Q1066487)

A subset \(X\subset S^ 3\) is called taut (resp. i-taut) in case the inclusion homomorphism in homology \(H_ j(X\cap B)\to H_ j(X)\) is injective for every round ball \(B\subset S^ 3\) and all j (resp. all \(j\leq i)\). Here it is proved that the only compact connected ANR taut sets in \(S^ 3\) are round spheres or cyclides of Dupin of horn or spindle type. Also the 1-taut sets in \(S^ 3\) are determined and a classification of the 0-taut sets is sketched.