nLab twisted ordinary cohomology

Idea

The twisted cohomology version of ordinary cohomology.

Related concepts

• twisted de Rham cohomology

• twisted ordinary differential cohomology

• twisted K-theory

• twisted Cohomotopy

• twisted Bredon cohomology (twisted ordinary equivariant cohomology)

References

Original articles:

• Kurt Reidemeister, Überdeckungen von Komplexen, Crelle 173 (1935), 164-173 [doi:10.1515/crll.1935.173.164]

• Norman Steenrod, Homology With Local Coefficients, Annals of Mathematics, Second Series, Vol. 44, No. 4 (Oct., 1943), pp. 610-627 (jstor:1969099)

Classification of twisted cohomology by fibrations/bundles of Eilenberg-MacLane spaces over the classifying space of a twisting group (universally: the fundamental group of the domain space):

• Samuel Gitler, Cohomology operations with local coefficients, American Journal of Mathematics 85 2 (1963) 156–188 [doi:10.2307/2373208]

• Yasumasa Hirashima, A note on cohomology with local coefficients, Osaka J. Math., 16 1 (1979) 219-231 &lbrackeuclid:ojm/1200771839]

• Paul Goerss, J. F. Jardine, p. 332 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) [doi:10.1007/978-3-0346-0189-4]

• M. Bullejos, E. Faro, M. A. García-Muñoz, Homotopy colimits and cohomology with local coefficients, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 44 no. 1 (2003), p. 63-80 [numdam:CTGDC_2003__44_1_63_0]

Review:

• Jonathan Rosenberg, Twisted cohomology, in Encyclopedia of Mathematical Physics 2nd ed, Elsevier (2024) [arXiv:2401.03966]

Discussion as a special case of a more general notion of twisted non-abelian cohomology by $\infty$-categorical hom-spaces in slice $\infty$-toposes:

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Rem. 4.22 in: Principal $\infty$-bundles – General Theory, Journal of Homotopy and Related Structures, 10 4 (2015) 749-801 [doi:10.1007/s40062-014-0083-6, arXiv:1207.0248]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, Ex. 2.32 in: The Character Map in Twisted Non-Abelian Cohomology [arXiv:2009.11909]