by the adjunction , or more precisely its restriction to abelian groups $$\xymatrix{ {{\mathbb{Z}}: {(\mbox{abelian groups})}} \ar@<.3ex>[r] &
{{(\mbox{commutative rings})}: {GL_{1}},} \ar@<.3ex>[l] }$$ we prove the following (see also [@May:rant]).

\[t-th-units-intro\] The functor ${gl_{1}}$ participates as the right adjoint in an adjunction $$\label{eq:33}
\xymatrix{ {{{\Sigma^{\infty}}_{+}}{\Omega^{\infty}}: \operatorname{ho}{(\mbox{$(-1)$-connected spectra})}}
\ar@<.3ex>[r] & {\operatorname{ho}{(\mbox{${E_{\infty}}$ ring spectra})}: {gl_{1}}} \ar@<.3ex>[l] }$$ which preserves the homotopy types of derived mapping spaces.

With this in place, in §\[sec:thom-spectra\] we recall and extend the obstruction theory of [@MQRT:ersers] for ${E_{\infty}}$ orientations. Let $R$ be an ${E_{\infty}}$ ring spectrum, and suppose that $b$ is a spectrum over $b{{gl_{1}}R} = \Sigma{{gl_{1}}R}.$ Let $p$ be the homotopy pull-back in the solid diagram $$\label{eq:13}
\xymatrix{ {{{gl_{1}}R}} \ar@{=}[r] \ar[d] & {{{gl_{1}}R}} \ar[d]
 \\
{p} \ar[r] \ar[d] \ar@{-->}[ur] & {e{{gl_{1}}R}{\simeq}{\ast}} \ar[d]
 \\
{b} \ar[r]^-{f} \ar@{-->}[ur] & {b{{gl_{1}}R}.}  }$$ Note that if $B={\Omega^{\infty}}b$ and $P={\Omega^{\infty}}p$, then after looping down we have a fibration sequence $$\label{eq:5}
  {{GL_{1}}R} \to P \to B\to B{{GL_{1}}R}.$$ Note also that an $E_{\infty}$ map $\varphi:R\to A$ gives a diagram $$\label{eq:13A}
\xymatrix{ {{{gl_{1}}R}} \ar[r] \ar[d] & {{{gl_{1}}A}} \ar[d]
 \\
{p} \ar[r] \ar[d] \ar@{-->}[ur] & {e{{gl_{1}}A}{\simeq}{\ast}} \ar[d]
 \\
{b} \ar[r]^-{\tilde{\varphi}\circ f} \ar@{-->}[ur] & {b{{gl_{1}}A},} }$$ where we write $\tilde{\varphi}:bgl_1 R\to bgl_1 A$ for the induced 