I am interested in computing the cobordism group of oriented manifolds $M$ of dimension 7 endowed with real vector bundles $N$ of rank 5 with the following conditions on the Siefel-Whitney classes:

$ w_1(N) = 0 \;, \quad w_2(N) = w_2(TM) \;, \quad w_5(N) = 0 \quad (1)$

The first equation says that $N$ is an orientable bundle. The second says that $TM \oplus N$ is spin. The third is equivalent to the vanishing of the Euler class of $N$.

So to paraphrase the above, I want to know if given an oriented 7-manifold $M$ endowed with a bundle $N$ satisfying (1), it is always possible to find an oriented 8-manifold $M'$ bounded by $M$ together with a bundle $N'$ satisfying (1) and extending $N$ to $M'$.

If we ask the same question disregarding the bundle $N$, the obstruction is given by the stable homotopy group $\pi_7(MSO)$, which can be shown to vanish using the results of C.T.C. Wall. So given a 7-dimensional oriented manifold, one can always find an 8-dimensional manifold bounded by the latter.

Now taking into account $N$ and ignoring the last two conditions in (1), I believe that the obstruction to finding a bordism is given by $[S, \Sigma^{-7}MSO \wedge \Sigma^5 BSO]$, where $S$ is the sphere spectrum, $\wedge$ the smash product and $[.,.]$ denotes the homotopy classes of maps. Already at this level I am not sure how to compute this group.

**Edit:** As $N$ has to be an actual bundle and not only a stable one, the bordism group should be in this case $[S, \Sigma^{-7}MSO \wedge BSO(5)]$.

Finally, one has to take into account the last two constraints of (1). To this end, I imagine that one should use the fact that there is a map from $\Sigma^{-7}MSO \wedge \Sigma^5 BSO$ into $\Sigma^2H\mathbb{Z}_2 \wedge \Sigma^5H\mathbb{Z}_2$, determined by $(w_2(N) - w_2(TM), w_5(N))$ and that the relevant spectrum is in some appropriate sense the kernel of this map.

Any hint about how to compute this cobordism group, or reference to similar computations in the literature would be greatly appreciated.

For people curious about it, the motivation to compute this group comes from the physics of the M5-brane. The worldvolume of the M5-brane is oriented and 6-dimensional. It is embedded in an 11-dimensional manifold $X$ which is spin, with normal bundle $N$. The spin condition on $X$ and the orientability of $M$ account first two conditions in (1). To compute the global gravitational anomalies of the effective field theory on the worldvolume, one has to consider mapping tori of the worldvolume, endowed with a vector bundle $N$ satisfying (1). And it turns out that the best way to express the anomaly is in terms of an 8-dimensional manifold bounded by the mapping torus. This is why knowing if such bounded manifolds exist is crucial.