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Each Morita–Mumford–Miller (MMM) class  e   n   assigns to each genus  g ⩾2 surface bundle Σ  g  → E  2 n +2 → M  2 n   an integer  e   n   #  ( E → M ):=〈  e   n  , [ M ]〉∈ℤ. We prove that when  n  is odd the number  e   n   #  ( E → M ) depends only on the diffeomorphism type of  E , not on  g ,  M  or the map  E → M . More generally, we prove that  e   n   #  ( E → M ) depends only on the cobordism class of  E . Recent work of Hatcher implies that this stronger statement is false when  n  is even. If  E → M  is a holomorphic fibering of complex manifolds, we show, that for every  n , the number  e   n   #  ( E → M ) only depends on the complex cobordism type of  E .  We give a general procedure to construct manifolds fibering as surface bundles in multiple ways, providing infinitely many examples to which our theorems apply. As an application of our results, we give a new proof of the rational case of a recent theorem of Giansiracusa–Tillmann [7, Theorem A] that the odd MMM classes  e  2 i −1  vanish for any surface bundle that bounds a handlebody bundle. We show how the MMM classes can be seen as obstructions to low‐genus fiberings. Finally, we discuss a number of open questions that arise from this work.

On the geometric nature of characteristic classes of surface bundles