Homotopy theory of dihedral and quaternionic sets
A map \(f:A\to B\) of \(O(2)\)-spaces is called a weak equivalence if the induced map of fixed points \(f^H:A^H\to B^H\) is a weak equivalence for all finite cyclic and dihedral subgroups \(H\subset O(2)\). It is shown that the resulting homotopy category is equivalent to the homotopy category of dihedral sets as introduced by \textit{Z. Fiedorowicz} and \textit{J.-L. Loday} in [Trans. Am. Math. Soc. 326, No. 1, 57-87 (1991; Zbl 0755.18005)]. There is a similar relation between Pin(2)-spaces (where Pin(2) is the normalizer of \(S^1\) in \(S^3)\) and quaternionic sets. The central part of the paper is the construction of suitable subdivisions of simplicial sets. The results are proved via \textit{D. Quillen}'s theorem in [Ann. Math. (2) 90, 205-295 (1969; Zbl 0191.53702)] on adjoint functors between model categories inducing an equivalence between their homotopy categories.
scientific article; zbMATH DE number 1436052