Meeting Notes 2025-04-15

This Week’s Finds

No finds this week.

 

Focus Session

Speaker: Jim
Title: Characteristic Classes via Cohomology: Stiefel–Whitney & Spin Structures

 

1. Overview

• Continued the cohomology series by applying ring–structure methods to characteristic classes of manifolds.
• Goals:
  1. Define Stiefel–Whitney classes in cohomology and compute examples.
  2. Explain how vanishing of certain classes enables spinor bundle construction via frame and spin bundles.

 

2. Main Content

1. Stiefel–Whitney Classes
   • Given a real vector bundle \(E\to M\), its total Stiefel–Whitney class is
     \(w(E)=1 + w_1(E) + w_2(E) + \cdots\) in \(H^*(M;\mathbb{Z}_2)\) via the cup product.
   • Defined by pulling back the generator of $H^1(\mathbb{R}P^\infty;\mathbb{Z}_2)$ along the classifying map \(M\to B\,O(n)\).
   • Key properties:
     • Natural under pullback: \(w(f^*E)=f^* w(E)\).
     • Whitney sum formula: \(w(E\oplus F)=w(E)\cup w(F)\).
   • Examples:
     • Tangent bundle of \(S^n\): nonzero \(w_n\), so nonorientable for even $n$.
     • Real projective space \(\mathbb{R}P^m\): \(w_1\) generates $H^1(\mathbb{R}P^m;\mathbb{Z}_2)$.
2. Spin Structures & Vanishing Criteria
   • A manifold \(M\) admits a spin structure iff \(w_1(TM)=0\) and \(w_2(TM)=0\) in \(H^*(M;\mathbb{Z}_2)\).
   • Interpretation: vanishing of \(w_1\) implies orientability; vanishing of \(w_2\) allows lift of $O(n)$–bundle to $Spin(n)$–bundle.
   • Construction: given a principal $O(n)$–bundle \(P\), existence of principal $Spin(n)$–bundle \(\widetilde P\) fitting into
     \(1\to\mathbb{Z}_2\to Spin(n)\to O(n)\to1\).
   • Connection to spinors: once \(\widetilde P\) exists, form spinor bundle via fundamental spin representation to define spinor fields.

 

3. Discussion Points

• Geometric meaning of \(w_2\) as obstruction to lifting frame bundle.
• Examples of nontrivial spin structures (e.g.\ on tori vs. nonorientable surfaces).
• Relation between cohomology ring operations and characteristic class computations.

 

Up Next

Next Talk:

• Joshua on the Chern–Weil Homomorphism and Chern Classes

 

References

1. Milnor & Stasheff, Characteristic Classes.
2. Hatcher, Algebraic Topology.
3. Stiefel–Whitney Class (Wikipedia)
4. Spin Structure (Wikipedia)