# Madsen-Tillmann spectrum $MTE$ of the group $E$ which is defined in Freed-Hopkins's paper

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!