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)H'(-; E) with coefficients in a special kind of spectrum EE related to a fibration p:EBp : 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 BB, which might also be called, in the older topological terminology, an ex-spectrum.

For any map f:KBf:K\to B and, for LK,L\subset K, a partial lift h:LEh:L\to E of ff, he constructs a single obstruction class Γ(f)H(K,L;E)\Gamma(f)\in H'(K, L ; E) to a full lift g:KE.g:K\to E.

E h g p L K f B \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)