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Results tagged with characteristic-classes
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user 8032
Cohomology classes associated to vector bundles. Includes Stiefel-Whitney classes, Chern classes, Pontryagin classes, and the Euler class.
4
votes
Accepted
Non-(stable)-triviality of the tautological bundles
For simplicity, let's take $\Bbb K = \Bbb R$.
By the bundle classification theorem, your question amounts to understanding whether the
inclusion map
$$
G_k(\Bbb R^N) \to \underset j{\text{colim }} \ …
- 18.5k
11
votes
topological type of smooth manifolds with prescribed homotopy type and pontryagin class
In the $1$-connected case, one may argue as follows:
Let $X$ be a closed $1$-connected smooth $n$-manifold, $n \ge 5$. The theory of the Spivak fibration shows that any homotopy equivalence $f: M^n …
- 18.5k
6
votes
Can one disjoin any submanifold in $\mathbb R^n$ from itself by a $C^{\infty}$-small isotopy?
There are lots of counter-examples.
Here's one:
The fibration $\text{SO}(3) \to \text{SO}(4) \to S^3$ splits, since it is the principal bundle of the tangent bundle of $S^3$, and the latter is paral …
- 18.5k
14
votes
Accepted
Characteristic classes for block bundles
I don't know where the results are written down in one place (perhaps in the book of Madsen and Milgram?), but see the the end of this post for a list of references.
In any case, here is a proof of …
- 18.5k