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• Tags
  • #geomtop
• Refs:
  • http://math.mit.edu/~mbehrens/18.906spring10/prin.pdf: #resources/notes
• Links:
  • Seiberg-Witten theory
  • Pin group

spin

• The questions on existence and classification of spin structures may be completely answered in terms of the Stiefel-Whitney classes.
• Spinnable (admits spin structure) implies admitting a \(\mathrm{Spin}^{{ \scriptscriptstyle \mathbf C} }\) structure, but not conversely
• Every closed oriented 4-manifold admits a \(\mathrm{Spin}^{{ \scriptscriptstyle \mathbf C} }\) structure
  • There are closed non-oriented manifolds that are not spinnable
• Every closed oriented is spinnable.

• Explanation of string structure by vanishing of characteristic classes, using the Whitehead tower. Essentially it all depends on \(\pi_* {\operatorname{O}}_n\):

Pasted image 20211117172353.png

Spin^c