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integral Stiefel-Whitney class

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Cohomology

cohomology

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Special notions

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Definition

The short exact sequence of abelian groups

02/20 0\to \mathbb{Z}\stackrel{\cdot2}{\to} \mathbb{Z}\to\mathbb{Z}/2\mathbb{Z}\to 0

induces a fiber sequence

B nB nB n/2B n+1 \cdots\to\mathbf{B}^n \mathbb{Z}\to \mathbf{B}^n\mathbb{Z}\to \mathbf{B}^n\mathbb{Z}/2\mathbb{Z}\to \mathbf{B}^{n+1}\mathbb{Z}\to \cdots

and so, for any object XX, a fiber sequence

H(X,B n)H(X,B n)H(X,B n/2)β 2H(X,B n+1) \cdots\to\mathbf{H}(X,\mathbf{B}^n \mathbb{Z})\to \mathbf{H}(X,\mathbf{B}^n\mathbb{Z})\to \mathbf{H}(X,\mathbf{B}^n\mathbb{Z}/2\mathbb{Z})\stackrel{\beta_2}{\to} \mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z})\to \cdots

of cocycle ∞-groupoid (with respect to any ambient (∞,1)-topos H\mathbf{H}, such as Top \simeq ∞Grpd), where β 2\beta_2 is the Bockstein morphism asociated with the multiplication by 2.

The image via β 2\beta_2 of the nn-th Stiefel-Whitney map w nH(X,B n/2)w_n\in \mathbf{H}(X,\mathbf{B}^n\mathbb{Z}/2\mathbb{Z}) in H(X,B n+1)\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z}) is called the (n+1)(n+1)st integral Stiefel-Whithey map and is denoted by W n+1W_{n+1}.

One usually uses the same symbol to denote the image of this characteristic map in cohomology (on connected components ) of W n+1W_{n+1} in H n+1(X;)=π 0H(X,B n+1)H^{n+1}(X;\mathbb{Z})=\pi_0\mathbf{H}(X,\mathbf{B}^{n+1}\mathbb{Z}), and calls this the (n+1)(n+1)-th integral Stiefel-Whitney class.

Examples

Third integral SW class

The third integral Stiefel-Whitney class W 3(TX)W_3(T X) of the tangent bundle of an oriented nn-dimensional manifold XX vanishes if and only if the second Stiefel-Whitney class w 2(TX)w_2(T X) is in the image of the reduction mod 2 morphism

H 2(X;)H 2(X;/2). H^2(X;\mathbb{Z})\to H^2(X;\mathbb{Z}/2\mathbb{Z}) \,.

Since H 2(X;)H^2(X;\mathbb{Z}) classifies isomorphism classes of U(1)U(1)-principal bundles over XX and W 3(TX)W_3(T X) is the obstruction to the existence of a spin^c structure on XX, we see that XX has a spin cspin^c structure if and only if there exists a principal U(1)U(1)-bundle on XX “killing” the second Stiefel-Whitney class of XX.

In particular, when w 2(TX)w_2(T X) is killed by the trivial U(1)U(1)-bundle, i.e., when w 2(TX)=0w_2(T X)=0, then XX has a spin structure.

The vanishing of the third integral SW class, hence spin^c-structure is the orientation condition in complex K-theory KUKU over oriented manifolds. In the context of string theory this is also known as the Freed-Witten anomaly cancellation condition.

Seventh integral SW class

Analogously, the vanishing of the seventh integral SW class is essentially the condition for orientation in second integral Morava K-theory.

In the context of string theory this is also known as the Diaconescu-Moore-Witten anomaly cancellation condition.

chromatic levelgeneralized cohomology theory / E-∞ ringobstruction to orientation in generalized cohomologygeneralized orientation/polarizationquantizationincarnation as quantum anomaly in higher gauge theory
1complex K-theory KUKUthird integral SW class W 3W_3spin^c-structureK-theoretic geometric quantizationFreed-Witten anomaly
2EO(n)Stiefel-Whitney class w 4w_4
2integral Morava K-theory K˜(2)\tilde K(2)seventh integral SW class W 7W_7Diaconescu-Moore-Witten anomaly in Kriz-Sati interpretation

Revised on June 17, 2013 17:32:34 by Urs Schreiber (82.113.106.57)