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Spinc structure
Special tangential structure From Wikipedia, the free encyclopedia
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In spin geometry, a spinc structure (or complex spin structure) is a special classifying map that can exist for orientable manifolds. Such manifolds are called spinc manifolds. C stands for the complex numbers, which are denoted 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.
This article's lead section may need to be rewritten. (March 2025) |
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Definition
Let be a -dimensional orientable manifold. Its tangent bundle is described by a classifying map into the classifying space of the special orthogonal group . It can factor over the map induced by the canonical projection on classifying spaces. In this case, the classifying map lifts to a continuous map into the classifying space of the spinc group , which is called spinc structure.[1]
Let denote the set of spinc structures on up to homotopy. The first unitary group is the second factor of the spinc group and using its classifying space , which is the infinite complex projective space and a model of the Eilenberg–MacLane space , there is a bijection:[2]
Due to the canonical projection , every spinc structure induces a principal -bundle or equvalently a complex line bundle.
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Properties
- Every spin structure induces a canonical spinc structure.[3][4] The reverse implication doesn't hold as the complex projective plane shows.
- Every spinc structure induces a canonical spinh structure. The reverse implication doesn't hold as the Wu manifold shows.[citation needed]
- An orientable manifold has a spinc structure iff its third integral Stiefel–Whitney class vanishes, hence is the image of the second ordinary Stiefel–Whitney class under the canonical map .[5]
- Every orientable smooth manifold with four or less dimensions has a spinc structure.[4]
- Every almost complex manifold has a spinc structure.[6][4]
The following properties hold more generally for the lift on the Lie group , with the particular case giving:
- If is a spinc manifold, then and are spinc manifolds.[7]
- If is a spin manifold, then is a spinc manifold iff is a spinc manifold.[7]
- If and are spinc manifolds of same dimension, then their connected sum is a spinc manifold.[8]
- The following conditions are equivalent:[9]
- is a spinc manifold.
- There is a real plane bundle , so that has a spin structure or equivalently .
- can be immersed in a spin manifold with two dimensions more.
- can be embedded in a spin manifold with two dimensions more.
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See also
Literature
- Blake Mellor (1995-09-18). "Spinc manifolds" (PDF).
- "Stable complex and Spinc-structures" (PDF).
- Liviu I. Nicolaescu. Notes on Seiberg-Witten Theory (PDF).
- Michael Albanese und Aleksandar Milivojević (2021). "Spinh and further generalisations of spin". Journal of Geometry and Physics. 164: 104–174. arXiv:2008.04934. doi:10.1016/j.geomphys.2022.104709.
References
External links
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