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?



Is the total space of Fiber bundle bordant to 0 if the fiber is null bordant?
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"


