Articles of characteristic classes

Stiefel-Whitney classes with Z-coefficients

Stiefel-Whitney classes are defined (for example, in Milnor’s Characteristic classes) as elements of the cohomology groups with $\mathbb{Z}_2$-coefficients as follows. $$w_i(E)=\phi^{-1} Sq^i(u)$$ where $E$ is an $n$-plane bundle over $B$, $E_0$ is the complement of the zero section, $Sq^i: H^*(E,E_0)\to H^{*+i}(E,E_0)$, $u\in H^n(E,E_0)$ is the Thom class and $\phi: H^*(B)\to H^{*+n}(E,E_0)$ is the Thom isomorphism […]

The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

In Milnor/Stasheff Characteristic classes on page 180 there is a statement (inside a proof) that I don’t fully understand. Let $\tilde{G}_k$ be the oriented real grassmanian, $e$ its Euler class: there is an exact sequence $$ H^i(\tilde{G}_n)\stackrel{\smile e}{\longrightarrow} H^{i+n}(\tilde{G}_n) \stackrel{\lambda}{\longrightarrow} H^{i+n} (\tilde{G}_{n-1})\to H^{i+1}(\tilde{G}_n)\to\ldots $$ that comes from the Gysin sequence in which $\tilde{E}_0$—complement of the […]

First Pontryagin class on real Grassmannian manifold?

I wonder if real Grassmannian manifold $SO(p+q)/SO(p) \times SO(q)$ have nontrivial first Pontryagin class? I only have physics background and know really little about characteristic class theory.

Are there any non-trivial group extensions of $SU(N)$?

Are there any non-trivial group extensions of $SU(N)$? If not, can one show/prove there are no non-trivial group extensions of $SU(N)$? It is possibly partial related to the homotopy group property. Or one can try to argue from the exact sequence. Proof/Show: Let us call $Q=SU(N)$. If the above claim is true, namely, we cannot […]

How to calculate characteristic classes of tensor products?

I was given the following as an execrise: Prove that the tensor product of complex line bundles over $X$ satisfies the following relationship: $$c_{1}(\otimes E_{i})=\sum c_{1}(E_{i})$$ It is not clear to me how to solve this via the defining axioms of Chern classes. My personal guess is this should hold for other characteristic classes except […]

Chern Classes and Stiefel-Whitney Classes

I’m trying to understand the relationship between Chern classes and Stiefel-Whitney classes, and I came upon this problem (14-E) in Milnor-Stasheff’s Characteristic Classes. We are asked to define the Stiefel-Whitney classes in the same way as was constructed for Chern classes, using mod 2 coefficients: $w_n(\xi) := e(\xi)$ mod 2 and $w_i(\xi):=(\pi_0^*)^{-1}w_i(\xi_0)$ for $i<n$, where […]

Calculate the Wu class from the Stiefel-Whitney class

The total Stiefel-Whitney class $w=1+w_1+w_2+\cdots$ is related to the total Wu class $u=1+u_1+u_2+\cdots$: The total Stiefel-Whitney class $w$ is the Steenrod square of the Wu class $u$: \begin{align} w=Sq(u),\ \ \ Sq=1+Sq^1+Sq^2 +\cdots . \end{align} The Wu classes can be defined through the Steenrod square (is this right? see nLab). $$ Sq^k(x) = \begin{cases} u_k […]

Compute the weights of a $(\mathbb C^*)^{m+1}$-action on $H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$

Let $\mathbb T:=(\mathbb C^{*})^{m+1}$ be the complex torus and suppose $\mathbb T$ acts on $\mathbb C^{m+1}$ diagonally as follows $$ (t_0, \cdots, t_m) : (x_0, \cdots, x_m) \mapsto (t_0x_0, \cdots, t_mx_m)$$ This action can induces a natural $\mathbb T$-action on $\mathcal O_{\mathbb P^m}(1)$ and also on $X:=H^0(\mathbb P^m, \mathcal O_{\mathbb P^m}(1))$, so the vector space […]

Prop 12.8 in Bott & Tu

Yo! This proposition in Bott & Tu have been haunting me for a year or so since I always have to come back to this book for references. More precisely, the second equality in Proposition 12.8 in page 135 of Differential Forms in Algebraic Topology have a weird assertion. More precisely, let $E \twoheadrightarrow X$ […]

Cohomology ring of Grassmannians

I’m reading a paper called An Additive Basis for the Cohomology of Real Grassmannians, which begins by making the following claim (paraphrasing): Let $w=1+w_1+ \ldots + w_m$ be the total Stiefel-Whitney class of the canonical $m$-plane bundle over $G_m(\mathbb{R}^{m+n})$ and let $\bar{w}=1+\bar{w_1}+\ldots+ \bar{w_n}$ be its dual. Then $H^\ast G_m (\mathbb{R}^{m+n})$ is the quotient of the […]