# Questions tagged [characteristic-classes]

Characteristic classes are invariants of bundles living in the cohomology of the base. The most common examples of characteristic classes are the Chern, Stiefel–Whitney, and Pontryagin classes.

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### $H^i\left(BO_m ; \mathbb{Z}_2\right) \cong H^i\left(B O_n ; \mathbb{Z}_2\right)$ for $i \leq m$ and $i \leq n.$

The notes I am reading say that groups in the title are isomorphic. Could someone explain to me why it is the case? Here by $BO_n$ I mean the infinite Grassmann manifold of $n$-dimensional subspaces. ...
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### How to compute Characteristic Classes from scratch and get which structure it capture

For our Differential Geometry II course we are strictly follow Differenetial Geometry, Loring W. Tu. Currently, I am reading chapter $5$, Vector bundles and Characteristic Classes. I feel the author ...
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### If Chern classes are only defined for vector bundles, why can a $U(1)$ principal bundle have associated Chern classes?

When people talk about Chern classes of $U(1)$ principal bundles, do they really mean the Chern class of the associated vector bundle? As far as I can understand, Chern classes exist only for complex ...
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### Euler class of a tensor product bundle

I want to prove the result: $$c_1(L\otimes L')=c_1(L)+c_1(L')$$ where $L,L'$ are complex line bundles and $c_1(L):=e(L_{\mathbb R})$. My definition of Euler class is given by the pullback by zero ...
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### Chern-Weil theory in terms of pullback along classifying map

Let $G$ be a semisimple real Lie group and $M$ be a smooth manifold. The map on cohomology $H^{\ast}(BG,\mathbb{C}) \rightarrow H^{\ast}(BT,\mathbb{C})\simeq \mathbb{C}[\mathfrak{h}]$ induced by the ...
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### Proving $c_1(L \otimes L') = c_1(L) + c_1(L')$ for line bundles.

I'm trying to prove this seemingly innocent result that for line bundles $L$and $L'$ the Chern class of the tensor product bundle is given by $$c_1(L \otimes L') = c_1(L) + c_1(L').$$ I managed to ...
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### Pullback bundle and Chern classes

The following is from Bott and Tu. If $f : M \to N$ is a map between two manifolds and $E$ is a complex bundle over $N$, then the pullback $f^{-1}E$ is a bundle over $M$. If the Chern classes of $E$ ...
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### K Group Determined by Chern Classes

Why is it that the complex reduced K group of $\mathbb{CP}^2$ is determined by Chern classes $c_1$ and $c_2$? I am aware of the fact that the cohomology ring of complex Grassmannians is generated by ...
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### Algebraic topology reference for someone well acquainted with algebra and differential geometry

I have spent the last two or so years making myself well rather well acquainted with the foundational aspects of differential topology/geometry. I have also spent the last year taking a two course ...
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### First Chern class $c_1$ of $U(1)$ bundle over $S^2$ v.s. $\pi_1(S^1)=\mathbb{Z}$
Let us compare the four statements: Consider the $U(1)$ fiber over the $S^2$. Such that this $S^1$ fiber over $S^2$ gives a first Chern class $c_1$ on the $S^2$ with $c_1=1$. Consider the $S^1$ ...