In Freed-Hopkins's paper, the group $E(d)$ is defined to be the subgroup of $O(d)\times\mathbb{Z}_4$ consisting of the pairs $(A,j)$ such that $\det A=j^2$, where $\mathbb{Z}_4=\{\pm1,\pm\sqrt{-1}\}$ is the multiplicative group of order 4.

Let $E$ be the colimit of $E(d)$, consider the Madsen-Tillmann spectrum $MTE$ of the group $E$, it is the colimit of $\Sigma^dMTE(d)$, where $\Sigma^dMTE(d) = Thom(BE(d);\mathbb{R}^d- V_d)$, where $V_d$ is the induced vector bundle (of dimension $d$) by the map $BE(d) \to BO(d)$. In other words, $MTE = Thom(BE;-V)$, where $V$ is the induced virtual bundle (of dimension 0) by the map $BE\to BO$.

My question: Can we decompose $MTE$ as the smash product of some familiar spectra?

Some known results:

$\bullet$ There is a short exact sequence of groups $$1\to SO\to E\to \mathbb{Z}_4\to1.$$

$\bullet$ There is a fibration of classifying spaces $$BE\to BO \stackrel{w_1^2}{\to}B^2\mathbb{Z}_2$$ which is obtained from the short exact sequence of groups $$1\to\mathbb{Z}_2\to E\to O\to1,$$ where $w_1$ is the first Stiefel-Whitney class of the virtual bundle over $BO$.

Thank you!


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