The phrase twisted cohomology was used by Larmore in

Larmore, Twisted cohomology theories and the single obstruction to lifting, Pac JM 41 (1972) 755-769

to describe cohomology$H\prime (-;E)$ with coefficients in a special kind of spectrum$E$ related to a fibration$p:E\to B$.

The result is what May and Sigurdsson (see references at twisted cohomology) call a parameterized spectrum, the “parameters” being the points of $B$, which might also be called, in the older topological terminology, an ex-spectrum.

For any map $f:K\to B$ and, for $L\subset K,$ a partial lift $h:L\to E$ of $f$, he constructs a single obstruction class $\Gamma (f)\in H\prime (K,L;E)$ to a full lift $g:K\to E.$

$$\begin{array}{ccccc}& & & \to & E\\ & {}^{h}\nearrow & & {}^{g}\nearrow & {\downarrow}^{p}\\ L& \hookrightarrow & K& \stackrel{f}{\to}& B\end{array}$$\array{
&& &\to& E
\\
&{}^h\nearrow& &{}^{g}\nearrow& \downarrow^p
\\
L &\hookrightarrow& K &\stackrel{f}{\to}& B
}

The vanishing of this obstruction is necessary for the existence of a lifting, but it is sufficient only in the usual stable range.

Notice that his cohomology with coefficients in a spectrum does not mean the sequence of cohomology groups with coefficients in the sequence of spaces constituting the spectrum, but rather a single group. He does explore the relation between his single obstruction and the classical obstructions.

Revised on August 18, 2009 06:57:20
by Urs Schreiber
(134.100.222.156)