Assume F is null bordant. Does it imply that the total space of fiber bundle $F\hookrightarrow E\to M$ is null bordant?
in particular what if $F$ is sphere?
Assume F is null bordant. Does it imply that the total space of fiber bundle $F\hookrightarrow E\to M$ is null bordant? in particular what if $F$ is sphere? 


There exist oriented surface bundles $E \to B$ on closed surfaces such that $E$ has nonzero signature (first found by Atiyah and Hirzebruch). Hence $E$ is not (oriented) nullbordant, even though base and fibre are nullbordant. Textbook reference: Morita, Geometry of characteristic classes. A lot of material on characteristic numbers of total spaces of fibre bundles is contained in Hirzebruch "Manifolds and modular forms" 

