In spin geometry, a spin<sup>c</sup> group (or complex spin group) is a Lie group obtained by the spin group through twisting with the first unitary group. C stands for the complex numbers, which are denoted . An important application of spin<sup>c</sup> groups is for spin<sup>c</sup> structures, which are central for SeibergâÂÂWitten theory.
The spin group is a double cover of the special orthogonal group , hence acts on it with . Furthermore, also acts on the first unitary group through the antipodal identification . The spin<sup>c</sup> group is then:
with . It is also denoted . Using the exceptional isomorphism , one also has with:
For all higher abelian homotopy groups, one has:
for .