# Tag Info

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### If two smooth manifolds are homeomorphic, then their stable tangent bundles are vector bundle isomorphic

The result you are hoping for is in fact false. In section 9 of Microbundles: Part I, Milnor constructs an open set $U \subset \mathbb{R}^m$. With its standard smooth structure, the (stable) tangent ...
• 17.3k

### The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

I'll use the definition of stack as a (weak) functor from the category of schemes to that of groupoids (as opposed to the definition as a fibered category over the category of schemes). The ...
• 40.4k
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• 33.4k
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### The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$

The category of maps from a test object $T$ to a quotient stack $[X/G]$ has the following general form. Objects are pairs $(P, f)$, where $P$ is a $G$-torsor over $T$, and $f: P \to X$ is a $G$-...
• 43.2k

### Vector bundles vs principal $G$-bundles

I believe you have answered this yourself. Principal bundles are trivial iff they admit a global smooth section. Vector bundles always admit a global smooth section (zero section). Therefore vector ...
Accepted

### Is every vector bundle over a noncompact finite-dimensional manifold a summand of a trivial bundle?

The proposition you refer to holds for any space homotopy equivalent to a finite dimensional CW complex. Here are the main points. The property that any vector bundle $E$ over a space has a ...
• 26.8k
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### Who discovered this definition of Stiefel-Whitney classes?

For the relation $Sq(U) = \Phi(w)$, where $Sq$ is the total Steenrod squaring operation, $U$ is the Thom class, $\Phi$ is the Thom isomorphism and $w$ is the total Stiefel-Whitney class, I would cite ...
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### When can we cancel vector bundles from tensor products?

According to this preprint, over a connected proper algebraic variety $X$ there is a universal reductive group $G$ such that isomorphism classes of vector bundles of rank $n$ are in bijection with ...
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• 17.3k
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### Fourth obstruction, Pontryagin and Euler class

Geometric generators for $\pi_3(SO(4))$ have been identified in §22 of Steenrod's "Topology of fibre bundles", using the identification of $S^3$ as unit quaternions. Conjugation of quaternions induces ...
• 16.6k