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Sep, 2019
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Twisted Cohomology

The idea of twisted K -theory can be traced back to the paper [ DK70 ] by Donovan and Karoubi. Its recent popularity is mainly due to the appearance of twisted K -theory in phypsics .

We also have twisted versions of generalized cohomology theories other than K -theory . The cohomology with local coefficients can be also regarded as a kind of twisted cohomology theory.

Atiyah and Segal briefly describe an idea of twisting generalized cohomology theories in [ AS04 ] . And they discuss twistings of de Rham cohomology in § 6 of [ AS06 ] .

There is an attempt of axiomatization of twisted generalized cohomology by Bunke and Schick [ BSa ] . They also discuss axioms of twisted cohomology theory for orbispaces in [ BSb ] .

Although there is a description of how to construct a twisted version of a generalized cohomology theory in the above mentioned paper by Atiyah and Segal, we should probably use parametrized spectra as is done by Waldmüller in [ Wal ] . He constructs spectra representing twisted K -theory and twisted Spin c -cobordism. See Part V of the book [ MS06 ] by May and Sigurdsson, for more details. There is also a description of equivariant versions.

Ando, Blumberg, and Gepner [ ABG + b ABGa ] consider twistings as Thom spectra (in a general sense).

Sati [ Sat ] suggests to use twistings of tmf for M-theory .

References