# Tagged Questions

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

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### Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
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### How can I equip the universal vector bundle over $\mathbf{G}(k,n)$ with a connection?

When constructing the grassmannians $\mathbf{G}_\mathbb{F}(k,n)$ for $\mathbb{F} = \{\mathbb{R},\mathbb{C} \}$ there is a natural vector bundle $$\mathbf{G}_{k,n} \to \mathbf{G}_\mathbb{F}(k,n)$$ ...
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### Cohomology of a classifying space

I would like some advice on the following problem: I have a topological group $G=\langle H,g\rangle$, where $H$ is a subgroup and $h$ is an element. Using a result of classifying spaces $Bi:BH \to BG$...
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### Is multiplication by a Stiefel-Whitney class an injective map?

I have a doubt: In cohomology, when you multiply by a Stiefel-Whitney class is it always an injective map? For example: is $$H^{j-1}(X)\xrightarrow{\smile\ w_1}H^{j}(X)$$ always injective? Thanks!
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### Construction of Grassmann manifolds

Is there a way to construct the Grassmann manifold via block matrices? For example the upper triangular matrices stabilize the (coordinate) basis of $\mathbb R^n$.
Let $\xi$ be an $n$-dimensional vector bundle over a manifold $M$ such that the reduced cohomology $\tilde H^*(M;\mathbb{Z})$ is entirely torsion (every element has finite order under addition). ...