In spin geometry, a spinc structure (or complex spin structure) is a generalization of a spin structure. In mathematics, these are used to describe spinor bundles and spinors, which in physics are used to describe spin, an intrinsic angular momentum of particles after which they have been named. Since spinc structures also exist under weakened conditions, which might not allow spin structures, they provide a suitable alternative for such situations. Orientable manifolds with a spinc structure are called spinc manifolds. C stands for the complex numbers, which are denoted C {\displaystyle \mathbb {C} } and appear in the definition of the underlying spinc group. In four dimensions, a spinc structure defines two complex plane bundles, which can be used to describe negative and positive chirality of spinors, for example in the Dirac equation of relativistic quantum field theory. Another central application is Seiberg–Witten theory, which uses them to study 4-manifolds.
Definition Let M {\displaystyle M} be a n {\displaystyle n} -dimensional orientable manifold. Its tangent bundle T M {\displaystyle TM} is described by a classifying map M → BSO ( n ) {\displaystyle M\rightarrow \operatorname {BSO} (n)} into the classifying space BSO ( n ) {\displaystyle \operatorname {BSO} (n)} of the special orthogonal group SO ( n ) {\displaystyle \operatorname {SO} (n)} . It can factor over the map BSpin c ( n ) → BSO ( n ) {\displaystyle \operatorname {BSpin} ^{\mathrm {c} }(n)\rightarrow \operatorname {BSO} (n)} induced by the canonical projection Spin c ( n ) ↠ SO ( n ) {\displaystyle \operatorname {Spin} ^{\mathrm {c} }(n)\twoheadrightarrow \operatorname {SO} (n)} on classifying spaces. In this case, the classifying map lifts to a continuous map M → BSpin c ( n ) {\displaystyle M\rightarrow \operatorname {BSpin} ^{\mathrm {c} }(n)} into the classifying space BSpin c ( n ) {\displaystyle \operatorname {BSpin} ^{\mathrm {c} }(n)} of the spinc group Spin c ( n ) {\displaystyle \operatorname {Spin} ^{\mathrm {c} }(n)} . Its homotopy class is called spinc structure. Assume M {\displaystyle M} has a spinc structure. Let then Spin c ( M ) {\displaystyle \operatorname {Spin} ^{\mathrm {c} }(M)} denote the set of spinc structures on M {\displaystyle M} . The first unitary group U ( 1 ) {\displaystyle \operatorname {U} (1)} is the second factor of the spinc group and using its classifying space BU ( 1 ) ≅ BSO ( 2 ) {\displaystyle \operatorname {BU} (1)\cong \operatorname {BSO} (2)} , which is the infinite complex projective space C P ∞ {\displaystyle \mathbb {C} P^{\infty }} and a model of the Eilenberg–MacLane space K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} , there is a bijection:
Spin c ( M ) ≅ [ M , BU ( 1 ) ] ≅ [ M , C P ∞ ] ≅ [ M , K ( Z , 2 ) ] ≅ H 2 ( M , Z ) . {\displaystyle \operatorname {Spin} ^{\mathrm {c} }(M)\cong [M,\operatorname {BU} (1)]\cong [M,\mathbb {C} P^{\infty }]\cong [M,K(\mathbb {Z} ,2)]\cong H^{2}(M,\mathbb {Z} ).}
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