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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.

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Xiao-Gang Wen
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    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
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    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
  • Are there a class of manifolds you are interested in? – Ryan Budney Oct 31 '14 at 20:32
  • orientable manifolds or orientable manifolds. – Xiao-Gang Wen Oct 31 '14 at 22:07
  • Uh huh. Okay, I'll think about it. – Ryan Budney Oct 31 '14 at 22:15
  • Sorry, I mean orientable manifolds or unorientable manifolds. – Xiao-Gang Wen Oct 31 '14 at 22:20
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    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 01 '14 at 05:37

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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.

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Alex Degtyarev
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    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
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    "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
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    "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
  • If $X$ is not orientable, in the above congruence you should change $w_2$ to $w_2+w_1^2$. – Alex Degtyarev Oct 31 '14 at 23:02
  • Alex: Thank you very much. You just showed that $w_2(X)c=c^2\bmod2$ is very general that applies to any $G$-bundle. Here I try to find all such relations. For example, in 5-dimension, do we have any relations between $w_1^3 c_1$, $w_3c_1$, $w_1 c_1 c_1$, and other pure SW classes ? Maybe in this case, there is no relation. – Xiao-Gang Wen Nov 01 '14 at 00:17
  • If you want to write equations in which $w_i$ is for the tangent bundle and $c_n$ for any bundle, then it would help to put this into the notation: $w_i(\tau_M)$ and $c_n(\xi)$ for example. – Robert Bruner Nov 01 '14 at 02:12
  • @Xiao-GangWen I edited my answer. – Alex Degtyarev Nov 01 '14 at 10:02
  • Alex: Thank you very much. One question, I know that the Wu formulas applies to SW classes $w_i$. But if I have a single $a \in H^i(X,Z_2)$, I do not know how to apply the Wu foormula. – Xiao-Gang Wen Nov 01 '14 at 11:00
  • I don't understand the question. If you have just a class, you have to deal with $\operatorname{Sq}$: there's no Wu formulas. If you have a $G$-bundle, you apply the Wu formulas to the SW classes of the underlying $O$-bundle: they are polynomials in the $G$-characteristic classes via the map $H^(BO;\Bbb Z_2)\to H^(BG;\Bbb Z_2)$ induced by the homomorphism $G\to O$. Essentially, all relations suggested are for the SW classes of the two bundles. – Alex Degtyarev Nov 01 '14 at 11:23
  • Of course, it may happen that $H^(BO;\Bbb Z_2)$ is essentially smaller than $H^(BG;\Bbb Z_2)$. In this case, you have to derive the appropriate "Wu formulas" yourself, by computing the Steenrod squares in $H^*(BG;\Bbb Z_2)$. – Alex Degtyarev Nov 01 '14 at 11:30
  • Alex: Thank you very much for the explaination. Let $G=U(1)$, we have $H^(BU(1),Z_2)$ generate by $c_1$ mod 2. Do you mean that we have $w_2(U(1))=c_1$ mod 2 and other $w_i(U(1))=0$. If we choose $G=Z_2$, we have $H^(BZ_2,Z_2)$ generate by $a_1$ in $H^1(BZ_2,Z_2)$. Do you mean that we have $w_1(Z_2)=a_1$ and other $w_i(Z_2)=0$. – Xiao-Gang Wen Nov 01 '14 at 12:30
  • $H^*(BU(1);\Bbb Z_2)=\Bbb Z_2[v_2]$, generated by $v_2:=c_1\bmod2$, and $\operatorname{Sq}^1v_2=0$, since it's an integral class. (Alternatively, one obviously has $v_1=0$; these $v_i$ are the SW classes of your $U(1)$-bundle.) As usual, $\operatorname{Sq}^2v_2=v_2^2$, and the Steenrod squares of powers $v_2^i$ are given by Cartan's formula. – Alex Degtyarev Nov 01 '14 at 12:32
  • Sorry, I replied before you edited. Yes, right, both for $U(1)$ and $\Bbb Z_2=O(1)$. – Alex Degtyarev Nov 01 '14 at 12:36
  • Oh thank you very much. That solve my problem. All I need to learn is "the SW classes $w_i(G)$ of the underlying O-bundle of the G-bundle: they are polynomials in the G-characteristic classes via the map $H^∗(BO;Z2)\to H^*(BG;Z2)$ induced by the homomorphism $G\to O$." There are many relations between $w_i(G)$'s and $w_i(O_X)$'s, where $w_i(O_X)$ are the SW classes of the tangent bundle of X. I hope the above summary is correct. Thank you very much. – Xiao-Gang Wen Nov 01 '14 at 12:47
  • FYI, the discussions here are usful for my paper http://arxiv.org/abs/1410.8477 – Xiao-Gang Wen Nov 01 '14 at 13:27
  • It is amazing, how classifying space $BG$ can be applied to theory in physics !? I thought, that it is too abstract notion. My professor Jackowski encouraged me to study those objects, but I gave up. –  Oct 18 '17 at 08:43