Does the unit sphere in a Banach space carries a structure of a CW complex? What about Finsler Manifolds?
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$\begingroup$ The unit sphere in $c_0$ is the completion of a "uniform polyhedron", arxiv.org/abs/1109.0346. The completion operation is quite harmless, for every uniform polyhedron $P$, in that there exists a self-homotopy $h_t$ of the completion of $P$ such that $h_0$ is the identity, and for each $t>0$, the image of $h_t$ lies in $P$. $\endgroup$– Sergey MelikhovCommented Jul 28, 2012 at 19:40
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The answer is no in infinite dimension (and yes in the finite dimensional case, of course). Because the sphere in a Banach space is Baire. But an infinite dimensional CW-complex is not, being a countable union of finite dimensional skeletons.
