Computing a cobordism group of manifolds endowed with a real vector bundle with constraints on the Stiefel-Whitney classes 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.  
 A: This is similar to András's answer, but a bit different...
Start with $BSO(5)$ and kill $w_5$, i.e take the fiber $F$ of
$w_5:BSO(5)\to K(Z_2,5)$. Now consider the composition
$\xi: BSpin \times F \to BSO\times BSO(5)\to BO\times BO \to BO$,
where the first two arrows are products of the obvious map and 
the last one corresponds to addition of vector bundles.
Finally take the Thom spectrum of the pullback of the universal 
bundle over BO along $\xi$.
The reason is that $N$ is classified by a map to $F$, and the 
difference $\nu-N$ is classified by a map to $BSpin$.
(I assume you want the spin structures on your manifolds and bordisms too.
Spin structures on $\nu-N$ are in bijection with spin structures on $TM+N$.)
Then the Thom spectrum is $MSpin\wedge \Sigma^{-5}TF$, where $TF$ is the
Thom space for $F\to BSO(5)$, so the bordism group is $\Omega_{12}^{Spin}(TF)$. One can try to compute this via the Atiyah-Hirzebruch spectral sequence,
or also by using the Adams spectral sequence.
A: Here is the space whose stable homotopy group will be the cobordism group you are looking for.
Start with $BSpin(5)$ and kill the cohomology class $w_5$ in  it.
(That is consider the map $w_5: BSpin(5) \to K(Z_2,5)$, pull back by this map the tautological
(path) fibration. Denote by $B \to BSpin(5)$ the obtained fibration. $B$ is the resulting space, which is obtained from $BSpin(5)$ after killing $w_5.)$
Pull back the 5-dimensional universal spin vector bundle from $BSpin(5)$ to $B.$
Let $T$ be the Thom space of this bundle. This is the space whose stable homotopy group is your cobordism group.
Namely $\pi_{q+12}(S^qT)$ is your cobordism group.
