nLab dihedral homology

Idea

The idea of cyclic homology can be generalized to other skew-simplicial groups as shown by Krasauskas and independently by Fiodorowicz. The simplest analogue is the dihedral homology.

There is also a geometric analogy where

• Hochschild homology is essentially the cohomology of free loop spaces;

• cyclic homology is essentially the cohomology of free loop spaces modulo the circle group $SO(2)$ action by rotating on the loops;

so dihedral homology is the cohomology of free loop spaces modulo the $O(2)$-action on loops (including reflections, as in the dihedral group).

References

• Massimiliano Ungheretti, Free loop spaces and dihedral homology (arXiv:1608.08140)
• T. Popelensky, On codihedral module for $\ast$-Hopf algebras, Journal of K-Theory, 2(2) (2008) 353–360 doi:10.1017/is008001018jkt033