nLab Larmore twisted cohomology

This entry is about particulars of the work of Larmore on cohomology with local coefficients (Steenrod, compare also Reidemeister earlier), a special case of what is now called twisted cohomology.

Larmore’s work

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 $H'(-; E)$ 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 $\Gamma(f)\in H'(K, L ; E)$ to a full lift $g:K\to E.$

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

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.