nLab
Wu class

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Wu classes are a type of universal characteristic class in 2\mathbb{Z}_2-cohomology that refine the Stiefel-Whitney classes.

Definition

For XX a topological space equipped with a class E:XBSO(n)E : X \to B SO(n) (a real vector bundle of some rank nn), write

w kH k(X, 2) w_k \in H^k(X, \mathbb{Z}_2)

for the Stiefel-Whitney classes of XX. Moreover, write

:H k(X, 2)×H l(X, 2)H k+l(X, 2) \cup : H^k(X, \mathbb{Z}_2) \times H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2)

for the cup product on 2\mathbb{Z}_2-cohomology groups and write

Sq k():H l(X, 2)H k+l(X, 2) Sq^k(-) : H^l(X, \mathbb{Z}_2) \to H^{k+l}(X, \mathbb{Z}_2)

for the Steenrod square operations.

Definition

The Wu class

ν kH k(X, 2) \nu_k \in H^k(X,\mathbb{Z}_2)

is defined to be the class that “represents” Sq k()Sq^k(-) under the cup product, in the sense that for all xH nk(X, 2)x \in H^{n-k}(X, \mathbb{Z}_2) where nn is the dimension of XX, we have

Sq k(x)=ν kx. Sq^k(x) = \nu_k \cup x \,.

(e.g. Milnor-Stasheff 74, p. 131-133)

Remark

In other words this says that the lifts of Wu classes to integral cohomology (integral Wu structures) are characteristic elements of the intersection product on integral cohomology, inducing quadratic refinements.

Properties

Relation to Stiefel-Whitney classes

The total Stiefel-Whitney class ww is the total Steenrod square of the total Wu class ν\nu.

w=Sq(ν). w = Sq(\nu) \,.

Solving this for the components of ν\nu in terms of the components of ww, one finds the first few Wu classes as polynomials in the Stiefel-Whitney classes as follows

  • ν 1=w 1\nu_1 = w_1;

  • ν 2=w 2+w 1 2\nu_2 = w_2 + w_1^2

  • ν 3=w 1w 2\nu_3 = w_1 w_2

  • ν 4=w 4+w 3w 1+w 2 2+w 1 4\nu_4 = w_4 + w_3 w_1 + w_2^2 + w_1^4

  • ν 5=w 4w 1+w 3w 1 2+w 2 2w 1+w 2w 1 3\nu_5 = w_4 w_1 + w_3 w_1^2 + w_2^2 w_1 + w_2 w_1^3

Relation to Pontryagin classes

Proposition

Let XX be an oriented manifold TX:XBSO(n)T X : X \to B SO(n) with spin structure T^X:XBSpin(n)\hat T X : X \to B Spin(n). Then the following classes in integral cohomology of XX, built from Pontryagin classes, coincide with Wu-classes under mod-2-reduction 2\mathbb{Z} \to \mathbb{Z}_2:

  • ν 4=12p 1\nu_4 = \frac{1}{2} p_1

  • ν 8=18(11p 1 220p 2)\nu_8 = \frac{1}{8}(11 p_1^2 - 20 p_2)

  • ν 12=116(37p 1 3100p 1p 2+80p 3)\nu_{12} = \frac{1}{16}(37 p_1^3 - 100 p_1 p_2 + 80 p_3).

(all products are cup products).

This is discussed in Hopkins-Singer, page 101.

Corollary

Suppose XX is 8 dimensional. Then, for GH 4(X,)G \in H^4(X, \mathbb{Z}) any integral 4-class, the expression

GGG12p 1H 4(X,) G \cup G - G \cup \frac{1}{2}p_1 \in H^4(X, \mathbb{Z})

is always even (divisible by 2).

Proof

By the basic properties of Steenrod squares, we have for the 4-class GG that

GG=Sq 4(G). G \cup G = Sq^4(G) \,.

By the definition 1 of Wu classes, the image of this integral class in 2\mathbb{Z}_2-coefficients equals the cup product with the Wu class

GGG12p 1=Sq 4(G)Gν 4=0mod2., G \cup G - G \cup \frac{1}{2}p_1 = Sq^4(G) - G \cup \nu_4 = 0 \; mod \; 2. \,,

where the first step is by prop. 1.

Applications

To higher dimensional Chern-Simons theory

Remark

The relation 1 plays a central role in the definition of the 7-dimensional Chern-Simons theory which is dual to the self-dual higher gauge theory on the M5-brane. In this context it was first pointed out in (Witten 1996) and later elaborated on in (Hopkins-Singer).

Specifically, in this context GG is the 4-class of the circle 3-bundle underlying the supergravity C-field, subject to the quantization condition

G 4=12(12p 1)+a, G_4 = \frac{1}{2}(\frac{1}{2}p_1) + a \,,

for some aH 4(X,)a \in H^4(X, \mathbb{Z}), which makes direct sense as an equation in H 4(X,)H^4(X, \mathbb{Z}) if the spin structure on XX happens to be such 12p 1\frac{1}{2}p_1 is further divisible by 2, and can be made sense of more generally in terms of twisted cohomology (which was suggested in (Witten 1996) and made precise sense of in (Hopkins-Singer) ).

For simplicity, assume that 12p 1\frac{1}{2}p_1 of XX is further divisible by 2 in the following. We then may consider direct refinements of the above ingredients to ordinary differential cohomology and so we consider differential cocycles a^,G^H^ 4(X)\hat a, \hat G \in \hat H^4(X) with

(1)G^=12(12p^ 1)+a^H^ 4(X), \hat G = \frac{1}{2}(\frac{1}{2}\hat \mathbf{p}_1) + \hat a \in \hat H^4(X) \,,

where the differential refinement 12p^ 1\frac{1}{2}\hat \mathbf{p}_1 is discussed in detail at differential string structure.

Now, after dimensional reduction on a 4-sphere, the action functional of 11-dimensional supergravity on the remaining 7-dimensional XX contains a higher Chern-Simons term which up to prefactors is of the form

G^expi X(G^G^(14p^ 1) 2), \hat G \mapsto \exp i \int_X ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,,

where

Using (1) this is

=expi X(a^a^+a^12p^ 1). \cdots = \exp i \int_X \left( \hat a \cup \hat a + \hat a \cup \frac{1}{2}\hat \mathbf{p}_1 \right) \,.

But by corollary 1 this is further divisible by 2. Hence the generator of the group of higher Chern-Simons action functionals is one half of this

G^expi X12(G^G^(14p^ 1) 2). \hat G \mapsto \exp i \int_X \frac{1}{2} ( \hat G \cup \hat G - (\frac{1}{4}\hat \mathbf{p}_1)^2 ) \,.

In (Witten 1996) it is discussed that the space of states of this “fractional” functional over a 6-dimensional Σ\Sigma is the space of conformal blocks of the self-dual higher gauge theory on the M5-brane.

References

The original reference is

  • Wen-Tsun Wu, On Pontrjagin classes: II Sientia Sinica 4 (1955) 455-490

See also around p. 228 of

and section 2 of

  • Yanghyun Byun, On vanishing of characteristic numbers in Poincaré complexes, Transactions of the AMS, vol 348, number 8 (1996) (pdf)

and

  • Robert Stong, Toshio Yoshida, Wu classes Proceedings of the American Mathematical Society Vol. 100, No. 2, (1987) (JSTOR)

Details are reviewed in appendix E of

This is based on or motivated from observations in

More discussion of Wu classes in this physical context is in

which also summarizes many standard properties of Wu classes.

Revised on June 14, 2014 11:34:35 by Anonymous Coward (99.121.56.222)