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Let $E$ be a vector bundle on a topological space $X$.Thanks to Allen Hatcher's book "Vector Bundles and K theory", the construction of sphere bundle $S(E)$ can be done without any inner product on fibers. It is a result of the equivalent relation on each punctured fiber: $v\sim w$ if $v=\lambda w$ for some positive scalar $\lambda$.

We define an equivalent relation on the set of all vector bundles over $X$ as follows: $E$ is equivalent to $F$ if their corresponfing sphere bundles $S(E)$ and $S(F)$ are isomorphic fiber bundles.

Is the above equivalent relation compatible to direct sum and tensor product of vector bundles in the following sense;

If $E_1,E_2$ are equivalent bundles and $F_1,F_2$ are also equivalent bundles in the above sense, are $E_1\oplus F_1, E_2 \oplus F_2$ are equivalent bundles? What about the tensor product, instead of direct sum? Can this equivalent relation define a kind of $K$ theory?

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    $\begingroup$ Maybe I am missing something here, but do you have an example where $E$ and $F$ are equivalent but not isomorphic as vector bundles? If there isn't such example, then it will exactly give $K$-theory. It is clear that if $E$ and $F$ are equivalent, then they are isomorphic as topological fibre bundles, since $E\simeq (S(E)\times[0,+\infty))/\sim$, where $(a,s)\sim(b,t)$ if $\pi(a)=\pi(b)$ and $s=t=0$. $\endgroup$
    – Pierre PC
    Commented Oct 14, 2020 at 19:55
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    $\begingroup$ @PierrePC There are examples, because $\text{Homeo}(S^{n-1}) \to O(n)$ is not a homotopy equivalence (the induced map on homotopy groups sometimes has kernel, sometimes has cokernel). I don't have explicit examples offhand. $\endgroup$
    – mme
    Commented Oct 14, 2020 at 21:12

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