Construction of Thom-Spectrum for G_2-Structures

Question

The motivation to this question is the paper of Crowley and Nordstrøm "A New Invariant of $G_2$-Structures". I am trying to find a homotopy theoretic interpretation of the following geometric situation: The exceptional Lie-group $G_2$ can be regarded as the stabilizer of a vector in $S^7$ under the action of $Spin(7)$ on $\mathbb{R}^8$ (this action can be depicted by octonionic multiplication). On a 7-dimensional $Spin$-manifold a $G_2$-structure on the tangent bundle $TM$ is equivalent to a homotopy class of non-vanishing unit sections $s\colon M \rightarrow \Sigma M$ into the associated real spinor bundle $\Sigma M$, giving an isomorphism of oriented Riemannian vector bundles $\underline{\mathbb{R}} \oplus TM \cong \Sigma M$. Crowley and Nordstrøm show, that every $G_2$-structure is bounded by a $Spin(7)$-structure, i. e. an 8-dimensional spin manifold $W$ with a unit section $\bar{s} \colon W \rightarrow \Sigma ^+ W$ into the positive half-spinor bundle, extending $s$. It follows that the Euler class of the positive half-spinor bundle vanishes.

Some thoughts so far have been: We are looking at sections into bundles associated to tangential structures, while bordism works in the stable normal direction. Thus, to fix a trivialization between normal and tangential structure, it could be worthwhile taking the homotopy fibre $B$ of the map $BG_2 \times BSpin(7) \rightarrow BSpin$ as an underlying space and construct the Thom-spectrum MB over B by the map $\xi \colon B \rightarrow BG_2 \times BSpin(7) \rightarrow BO(7) \rightarrow BO$. But if every $G_2$-structure is bounded by a $Spin(7)$-structure then the group $\pi _7 (MB)$ should vanish, correct? Another idea would be the classifying space $BG_2$ and its Thomspectrum. At least it is known that the bordism group is $\Omega ^{G_2}_7 \cong \mathbb{Z}/3\mathbb{Z}$. But in that situation I do not see how to transfer this normal $G_2$-structure to a tangential structure.

So my question is: Which choice of underlying space and Thomspectrum seems reasonable to deal with the explained geometric situation?

Answer 1

The second version of the arXiv post of the Crowley-Nordström paper goes into some detail in giving an answer to this question: see Section 7 of version 2. The out-line is as follows:

1) First define stable $G_2$-structures and stable tangential $G_2$-bordism.

2) Then make the standard identification of stable tangential bordism and normal bordism.

3) Finally, use the Thom spectrum for normal $G_2$ bordism - this is the suspension spectrum of the Thom space of the inverse to the canonical vector bundle $V$ over $BG$, where $V$ is obtained from the standard representation of $G_2$ on $\mathbb{R}^7$. i.e. our candidate for $MG_2$ is the suspension spectrum of $T(-V)$, where $-V$ is the K-theory inverse of $V$ and $T(-V)$ is its Thom space: of course, you have to take some care to make the rank of the representative of $-V$ and the indexing of the suspension spectrum match up.

Whether this is what you want, I am not sure, but did allows us to compare $G_2$ bordism and $SU(2)$ bordism and to give a geometric interpretation of the $\nu$-invariant modulo 3.