Dihedral homology and cohomology. Basic notions and constructions

The subject of the article is the foundations of the theory of dihedral homology and cohomology, the Hermitian analogue of cyclic homology and Connes–Tsygan cohomology. The article consists of four sections. In § 1 the notion of a dihedral object in a category is defined, and algebraic and homotopy properties of dihedral objects are studied. A detailed study is made of dihedral modules, i.e., dihedral objects in the category of modules over a commutative ring. § 2 contains several equivalent definitions of dihedral homology and cohomology of dihedral modules. One of them, in terms of derived functors, is convenient for obtaining general theorems on dihedral (co)homology; two others allow one to create effective means to compute this (co)homology. § 3 deals with establishing numerous connections between dihedral homology and other homology functors such as, say, Hochschild homology or cyclic homology. In § 4 the dihedral Chern character is introduced, and relations between Hermitian $K$-theory and dihedral homology are studied.
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