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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.
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
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