Oriented manifolds that fiber over 𝑆⁴

Abstract

Necessary and sufficient conditions are given for an oriented manifold M M to fiber up to cobordism over the 4 4 -sphere S 4 {S^4} (i.e. for M M to be oriented cobordant to a fiber bundle over S 4 {S^4} ). The result here extends those previously obtained for fiberings over S 1 {S^1} and S 2 {S^2} . In addition, fiberings over products of surfaces are studied with complete solutions (in the sense above) being given in most cases including those of S 2 × S 2 {S^2} \times {S^2} and ( S 1 ) 4 {({S^1})^4} .