# Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that $(B(V),S(V))$ cannot be obtained as a relative CW-complex in an "obvious" way.

I suspect however that B(V) that can be built up from S(V) by attaching cells (in no particular order). The absolute case is a lemma in hatchers k-theory book, but i am not sure if the inductive proof over the cells of $X$ works in the relative case, since checking this formally seem extremly tedious. But maybe someone here has already dealt with a similar issue.

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