Which bordism classes fiber over the circle?

Question

Let $\mathcal{G}$ denote a (stable) tangential structure such as $O$, $SO$, $Spin$, or $Pin^\pm$. Which bordism classes $[M,f]\in\Omega_*^\mathcal{G}(X)$ are represented by an $f:M\rightarrow X$ where the $\mathcal{G}$-manifold $M$ fibers over $S^1$?

Answer 1

A manifold $M$ fibres over $S^1$ with fibre $F$ if and only if it is isomorphic to the mapping torus $$T(h)=F \times [0,1]/\{(x,0) \sim (h(x),1)\vert x \in F\}$$ of an automorphism $h:F \to F$. Mapping tori are particular examples of open books. Walter Neumann's result for $G=SO$ extends to arbitrary $X$ with the signature replaced by the invariant of

Quinn, Frank, Open book decompositions, and the bordism of automorphisms. Topology 18 (1979), no. 1, 55-73

For odd-dimensional $M$ the invariant is 0. For even-dimensional $M$ the invariant is the asymmetric Witt class of the $Z[\pi_1(X)]$-module chain complex $C(\tilde{M})$ with Poincar\'e duality, where $\tilde{M}$ is the pullback to $M$ of the universal cover $\tilde{X}$ of $X$ - see Chapters 29,30 of

Ranicki, Andrew, High-dimensional knot theory, Springer Monographs in Mathematics, 1998.

Answer 2

The case $G=SO$ and $X=*$ is considered in

Neumann, Walter D., Fibering over the circle within a bordism class. Math. Ann. 192 1971 191–192.

where it is shown that a bordism class fibres over the circle if and only if it has signature zero.