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I've heard about the following result: for each two natural numbers $d,n\in\mathbb{N}$ one can find a number $k\in\mathbb{N}$ with the following property: for each CW-complex $X$ with $\dim X\leq d$ and a rank $n$ vector bundle $E\to X$ there is a continuos map into the Grassmanian $f:X\to G_n(\mathbb{R}^{n+k})$ such that $E\simeq f^*\gamma_n\left(\mathbb{R}^{n+k}\right)$ where $\gamma_n\left(\mathbb{R}^{n+k}\right)$ is the tautological rank $n$ vector bundle over $G_n(\mathbb{R}^{n+k})$.

Question 1 Could anybody give me reference to this theorem such that in the proof it is apparent how $k$ depends of $n,d$?

Question 2 if $n,k$ are given numbers, how many trivializations are needed for covering $\gamma_n\left(\mathbb{R}^{n+k}\right)$?

Of course, please let me know if the above formulation needs some corrections.

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  • $\begingroup$ I think usually they require in addition that $X$ should be compact; otherwise, $K$-theory doesn't work quite well. $\endgroup$ Oct 11, 2014 at 20:11
  • $\begingroup$ I think this is a combination of the representation result (which shows that on a paracompact Hausdorff space every rank n bundle is pullback of the universal one along the classifying map into the infinite Grassmannian) and a stabilization result (which, using obstruction theory, allows to get to a finite level). I would need to check the numbers for question 1. $\endgroup$ Oct 11, 2014 at 20:18
  • $\begingroup$ Maybe you could check Husemoeller's book on fiber bundles to find the answers to your question... $\endgroup$ Oct 11, 2014 at 20:21
  • $\begingroup$ Theorem 5.5 of Husemoller's book gives $k = n(m-1)$ where $m$ is the number of open sets in a covering of $X$ such that $E$ is trivial on all of them (if such a cover exists, in which case $E$ is said to be of finite-type over $X$). By Proposition 5.8, such an isomorphism of vector bundles occurs if and only if $E$ is of finite-type over $X$. Thus you'll want the compactness assumption on $X$ for your result (then all vector bundles are vacuously of finite-type). $\endgroup$ Oct 13, 2014 at 0:12
  • $\begingroup$ Well, it was my original motivation to pose my questiion: namely, I wanted to know how many open sets do I need to cover any at most $d$-dimensional CW complex (yes, I had to assume compactness) in such a way that for any rank $n$ vector bundle $E$ the restriction to these sets will be trivial. Therefore I need to know how to express $k$ by $d$ and $n$ and knowing this, how many open sets do I need to cover Grassmanian in such a way that the restrictions of tautological bunble will be trivial. In other words, I would like to find universal $k$. $\endgroup$
    – truebaran
    Oct 13, 2014 at 1:49

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