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history of cohomology with local coefficients

The subject of cohomology with local coefficients (twisted cohomology) begins with Reidemeister in his study of lens spaces in 1938 in the article Topologie der Polyeder und kombinatorischen Topologie der Komplexe. He considered the cohomology of a polyhedron (there called a manifold – such terminology hadn’t yet settled down and so is probably best considered as a combinatorial manifold) $M$ in terms of cochains of the universal covering space $U$ together with the action of the fundamental group of $M$ as deck transformations. His terminology was Überdeckung (covering space). Interestingly, this approach underlies the crossed complexes studied extensively by Ronnie Brown, Phil Higgins and others in the past four decades.

Zoran Škoda J. Nielsen. ¨Uber die Minimalzahl der Fixpunkte bei den Abbildungstypen der Ringfl¨achen. Math. Ann., 82(1-2):83–93, 1920. is probably the reference responsible for so called Nielsen invariant. Nielsen invariant and Reidemeister torsion are rather related (there are many article now on this relation, for example by A. Felshtyn); there are also some precursors like dilogarithm functions from the end of 19th century, closely related to the subject. Finally the first article having homological algebra by Cayley from 1840-s is claimed by Gelfand-Kapranov-Yale to use at one point an invariant which is in the calculations with Koszul complexes the analogue of the Whitehead torsion, but I was not able to do the comparison. There is a book

• J.L. Dupont, Scissors congruences, group homology and characteristic classes (.ps) , Lecture notes from Nankai Institute of Mathematics, 1998, Manuscript, University of Aarhus, Department of Math., Aarhus, 1999, 122 pp., and in: Nankai Tracts in Mathematics 1, World Scientific, Singapore, 2001.

which explains the relations of dilogarithms, Reidemeister torsion, Dehn invariant (which is from 1900) and its role in understanding the 19th century scissors congruence…The subject is very active now. After physicist Anatole Kirillov found the quantum dilogarithm and L. D. Fadeeev and Kashaev published a paper about it (and discovered the pentagon relation), this became an important topic in quantization (Fadeev’s modular double, quantization of Teichmuller spaces, work of A. Fock, L. Takhdajan, Aldrovandi etc.) of hyperbolic spaces and generally the hyperbolic geometry and with relations, together with classical dilogarithm to many other subjects (e.g. to the work of Reznikov and theory or regulators, specially Borel regulator in higher algebraic K-theory, important for study of motives (Goncharov et al.) and, as W. Nahm shown, also relevant for understanding some phenomena in CFT).

Steenrod in 1942 independently generalized the concept of ordinary cohomology to cohomology with coefficients in a local system of groups, later known as cohomology with local coefficients. In 1943, he acknowledged Reidemeister’s precedence showing that (after a suitable interpretation) the theory includes Reidemeister’s concept of Überdeckung.

In 1947, Eilenberg showed that the cohomology theory with a local system of groups is naturally equivalent to a theory of equivariants with respect to the deck transformation?s of cohomology with ordinary coefficients on the corresponding universal covering space.

In 1950, Olum extended Eilenberg’s obstruction? theory to the non-simple case, i.e. when the fundamental group acts non-trivially on the higher homotopy groups involved.

• Obstructions to extensions and homotopies, Annals 52 (1950) 1–52

In the 1960s and early 1970s, there were several papers addressing operations on cohomology with local coefficients:

• 1963, Gitler (Steenrod mod $p$), Cohomology operations with twisted coefficients, AJM 85 (1963)156–188

• 1966, McClendon, thesis – summarized in

• 1967, Emery Thomas, tc ops

• 1967, Larmore, tc ops

• 1969, McClendon, Higher order twisted cohomology operations, Invent. Math. 7 (1969) 183–214

• 1969, Larmore, tc

• 1970, Peterson, tc ops

• 1971, McClendon, tc ops

In 1972, the phrase twisted cohomology was used by Larmore to describe $H(-;ℰ)$, cohomology with coefficients in a special kind of spectrum $ℰ$ related to a fibration $p:E\to B.$ The result is what May and Sigurdsson call a parameterized spectrum, the parameters being the points of $B$, which might also be called, in the older topological terminology, an ex-spectrum.

In recent years, especially after the invention of twisted K-theory, ‘twisted cohomology’ has become the generic term, including not only twisted generalized (Eilenberg–Steenrod) cohomology but even what is known as twisted non-abelian cohomolgy. Among other things, non-abelian refers to the fact that no spectrum need be involved, but only a single target space, preferably at least a loop space.

Also in 1972 Robinson constructed Moore–Postnikov systems for non-simple fibrations. In particular, he provided twisted $K(\pi ,n)$s corresponding to cohomology with local coefficients.