# Relations between characteristic classes of a group and the Stiefel-Whitney/Pontryagin classes

Let $X$ be a closed manifold and $BG$ be the classifying space of a group $G$ A map from $X$ to $BG$ induce a map from $H^*(BG,Z)$ to $H^*(X,Z)$ by pull back. Let $GH^*(X,Z)$ be the subgroup of $H^*(X,Z)$ formed by the images of the above map $H^*(BG,Z)\to H^*(X,Z)$ for all the maps $X\to BG$.

In this case $a \in GH^*(X,Z)$ and the Stiefel-Whitney/Pontryagin classes may have some relations that are indenependent of $X$. What are those relations?

If we choose $G=U(1)$, this question bacomes: what are the relations (all the $X$ independent relations) between the Chern classes of a $U(1)$ bundle on $X$ and the Stiefel-Whitney/Pontryagin classes on $X$.

In question Relations between Stiefel-Whitney classes, the relations between Stiefel-Whitney classes on any $X$ are discussed.

== Added == This may be the better way to phrase the question: What are relations between the characteristic classes of the tangent bundle of a manifold and characteristic classes of an arbitrary $G$-bundles on the same manifold.

I think in 4-dimension and for $G=U(1)$, one of the relation is $(w_2+w_1^2) c^{U(1)}_1 = c^{U(1)}_1c^{U(1)}_1$ mod 2.

• I'm not sure if your question has been worked out by someone in that generality but presumably for "reasonable" collections of groups $G$ the answer is accessible. Do you have a few groups you're particularly concerned about? Reading your comment in Degtyarev's answer, depending on your manifold and the other bundles you're interested there may be more or less in the way of relations. I'd like to encourage you to give us a more concrete family of bundles you are interested in. I suspect working out one case might give you an idea for how to approach other cases. – Ryan Budney Oct 31 '14 at 15:52
• I am interested in some very simple groups, such as $Z_n$, $U(1)$, and $Z_2^T$. Here $Z_2^T$ is really the $Z_2$ group. The superscript $T$ means that $Z_2^T$ has a non-trivial action on the coefficient $Z\to -Z$. Certainly, a general result will be even better. – Xiao-Gang Wen Oct 31 '14 at 20:18
• For the $U(1)$ case, I wonder if there is a relation like $w_2 c^{U(1)}_1=(c^{U(1)}_1 )^2$ mod 2 in 4-dimensions. – Xiao-Gang Wen Oct 31 '14 at 20:22
• With the tangent bundle, the main relations among the Stiefel-Whitney classes are induced by Poincare duality. There's also relations coming from the structure of the cohomology of the classifying space. For arbitrary bundles the only relations you will get is the ones coming from the cohomology of the bundles. There's basically just one other restriction, given your space $X$, there's only certain maps $X \to BG$ that the homotopy-types of $X$ and $BG$ will allow. I'll give your examples some thought and reply if someone else does not, later this weekend. Have some trick-or-treating to do – Ryan Budney Oct 31 '14 at 20:28
• I don't see any good reason for relations of this form to exist. Can you explain the motivation for the question? – Qiaochu Yuan Nov 1 '14 at 5:37

## 1 Answer

There's no relations between SW, Pontrjagin, and Chern classes (within each set). On the other hand, for a $U(1)$-bundle, one has $w_1=0$ and $w_2=c_1\bmod2$. Such a bundle has no (rational) Pontrjagin classes.

More general setting Here is another bunch of relations; most likely these are all, but I'm not 100% certain. Let $\dim X=n$, and let $u_i:=u_i(X)$ be the $i$-th Wu class of $X$. (Recall that the total class is $u=\operatorname{Sq}^{-1}w(X)$, and $u_i$ are the homogeneous components.) Then, for any $a\in H^{n-i}(X;\Bbb Z_2)$, one has $u_ia=\operatorname{Sq}^ia$ in $H^n(X)=\Bbb Z_2$. This gives us relations $u_i=0$ for $i>n$, and these generate the ideal of all relations among the SW classes $w_*(X)$, see Relations between Stiefel-Whitney classes. Now, if one takes for $a_j\in H^j(X;\Bbb Z_2)$ the characteristic classes (SW, Chern, Pontrjagin, or such) of another bundle (or combinations thereof), then, using Wu formulas to express $\operatorname{Sq}^{n-j}a_j$ in terms of the $a_k$'s and substituting to the above formulas, one gets a bunch of relations between $w_*(X)$ and $a_*$.

The relation $(w_1^2(X)+w_2(X)) v_2=v_2^2$ mentioned in the comments is one of them. (Here, $v_2\in H^2(X;\Bbb Z_2)$ is any class, which, if desired, can be interpreted as $w_2$ or $c_1$ or whatever of another bundle.) Another one for a $4$-manifold is $w_1(X)v_3=v_1v_3$, where $v_i$ are the SW classes of another bundle.

• I think you misread the question--as I read it, it is asking about relations between the characteristic classes of the tangent bundle of a manifold and characteristic classes of arbitrary $G$-bundles. – Eric Wofsey Oct 31 '14 at 15:21
• Thank you Eric. I am a physicist. I did not phase my question properly. Indeed, I am asking about the relations between the characteristic classes of the tangent bundle of a manifold and characteristic classes of arbitrary G-bundles on the same manifold. Alex: $w_2=c_1$ mod 2 is the kind of relation that I am asking. I know that the Chern class $c_1$ of the tangent bundle of a $complex$ manifold has the relation $w_2=c_1$ mod 2. But I do not know if the Chern class $c^{U(1)}_1$ of any $U(1)$ bundle has the relation $w_2 = c^{U(1)}_1$ mod 2? – Xiao-Gang Wen Oct 31 '14 at 15:46
• "Indeed, I am asking about the relations between the characteristic classes of the tangent bundle of a manifold and characteristic classes of arbitrary G-bundles on the same manifold" To me, this seems contradictory: no relation whatsoever, especially if one wants one "independent of $X$". Or are you speaking about $G$-reduction of the tangent bundle? The relation $w_{2n}=c_n\bmod2$ holds for any $U(n)$-bundle. For other (in fact, all) universal relations between $c_i\bmod2$ and $w_j$ of the underlying real bundle just use the splitting principle. – Alex Degtyarev Oct 31 '14 at 18:05
• Here $w_i$ is for the tangent bundle, and $c_n$ is for an independent $U(1)$ bundle, which is not related to the tangent bundle. The relation should not depend on the topology of $X$ but may depend on the dimension of $X$. I think in 4-dimension we should have $w_2c_1=c_1^2$ mod 2. – Xiao-Gang Wen Oct 31 '14 at 22:04
• "Here wi is for the tangent bundle, and cn is for an independent U(1) bundle." Then, how can they be related? Of course, if $X$ is closed, oriented, and of dimension $4$, then $w_2(X)c=c^2\bmod2$ for any class $c\in H_2(X)/\operatorname{Tors}$, just because $w_2$ is the characteristic class of the intersection index form (one of the Wu formulas). But this has nothing to do with a $U(1)$-bundle (even though any class in $H^2$ is $c_1$ of such a bundle). – Alex Degtyarev Oct 31 '14 at 22:58