a map $$g: b\to b{{gl_{1}}S},$$ so that $$G = {\Omega^{\infty}}g: B \to B{{GL_{1}}S}$$ classifies a stable spherical fibration. Using Definition \[def-5\], we can form the ${E_{\infty}}$ Thom spectrum $Mg.$ In §\[sec:comp-thom-spectra\], we show that the spectrum underlying $Mg$ is the usual Thom spectrum of the spherical fibration classified by $G,$ as constructed for example in [@LMS:esht].

Now suppose that $R$ is an ${E_{\infty}}$ spectrum with unit $\iota: S\to
R$, and let $$f = b{{gl_{1}}\iota}\circ  g: b \to b{{gl_{1}}S} \to b{{gl_{1}}R}.$$ Then $$Mf {\simeq}Mg {\wedge}^{L} R,$$ and so we have an equivalence of derived mapping spaces $${{\mathscr{S}}[{E_{\infty}}]}(Mg,R) {\simeq}{(\mbox{${E_{\infty}}$ $R$-algebras})} (Mf,R).$$ If we let $b (S,R)$ be the pull-back in the solid diagram $$\label{eq:34}
\xymatrix{ {p} \ar[r] \ar[d] & {b (S,R)} \ar[r] \ar[d] & {{\ast}}
\ar[d]
\\
{b} \ar[r] \ar@{-->}[ur] & {b{{gl_{1}}S}} \ar[r] & {b{{gl_{1}}R},} }$$ then Theorem \[t-th-Thom-iso\] specializes to a result of May, Quinn, Ray, and Tornehave [@MQRT:ersers].

\[t-thom-iso-mqr\] The derived space of ${E_{\infty}}$ maps $Mg\to R$ is weakly equivalent to the derived space of lifts in the diagram .

The space of units and orientations {#sec:space-units-orient}
-----------------------------------

The authors of [@MQRT:ersers] describe ${E_{\infty}}$ orientations for ${E_{\infty}}$ ring spectra, and they also discuss not-necessarily ${E_{\infty}}$ orientations of ${E_{\infty}}$ spectra. As 