×

zbMATH — the first resource for mathematics

On \(S^3\)-equivariant homology. (English) Zbl 0988.19002
This paper describes connections between quaternionic homology and \(S^3\)-equivariant homology, where \(S^3\) is the group of quaternions of norm one. Quaternionic homology is constructed from the sequence of finite quaternionic groups together with some additional structure making them into what is called a crossed simplicial group. The theory of crossed simplicial groups shows that quaternionic homology is actually \(\text{Pin}(2)\)-equivariant homology, where \(\text{Pin}(2)\) is the normaliser of a maximal torus in \(S^3\). The author shows that \(S^3\)-equivariant homology cannot be associated to a crossed simplicial group, but he gives an exact sequence connecting it with \(\text{Pin}(2)\)-equivariant homology.
MSC:
19G38 Hermitian \(K\)-theory, relations with \(K\)-theory of rings
55N91 Equivariant homology and cohomology in algebraic topology
55T05 General theory of spectral sequences in algebraic topology
22E40 Discrete subgroups of Lie groups
PDF BibTeX XML Cite
Full Text: DOI EuDML