Spin Structure - Spinc Structures

Spinc Structures

A spinc structure is analogous to a spin structure on an oriented Riemannian manifold, but uses the spinc group, which is defined instead by the exact sequence

To motivate this, suppose that κ: Spin(n) → U(N) is a complex spinor representation. The center of U(N) consists of the diagonal elements coming from the inclusion i: U(1) → U(N), i.e., the scalar multiples of the identity. Thus there is a homomorphism

This will always have the element (-1,-1) in the kernel. Taking the quotient modulo this element gives the group SpinC(n). This is the twisted product

where U(1) = SO(2) = S1. In other words, the group Spinc(n) is a central extension of SO(n) by S1.

Viewed another way, Spinc(n) is the quotient group obtained from Spin(n) × Spin(2) with respect to the normal Z₂ which is generated by the pair of covering transformations for the bundles Spin(n) → SO(n) and Spin(2) → SO(2) respectively. This makes the spinc group both a bundle over the circle with fibre Spin(n), and a bundle over SO(n) with fibre a circle.

The fundamental group π₁(SpinC(n)) is isomorphic to Z.

If the manifold has a cell decomposition or a triangulation, a spinc structure can be equivalently thought of as a homotopy class of complex structure over the 2-skeleton that extends over the 3-skeleton. Similarly to the case of spin structures, one takes a Whitney sum with a trivial line bundle if the manifold is odd dimensional.

Yet another definition is that a spinc structure on a manifold N is a complex line bundle L over N together with a spin structure on TN ⊕ L.