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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}      .

Oriented manifolds that fiber over 𝑆⁴