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My last question (Is such a map always null-homotopic?) is quite similar. If you do not care about my motivation for these questions, you can skip to the last line. I asked if some assumptions were sufficient conditions to guarantee the non-existence of "phantoms" (essential maps that are non-essential on every finite sub-complex). Now I have thought a little longer and have a new extra condition to add which is true for the application I had in mind. In particular I want to prove the following

Conjecture: If $V\rightarrow M$ is a finite-dimensional real smooth vector bundle over a smooth $n$-manifold $M$ (possibly with boundary), then $V$ is trivial if and only if it is trivial restricted to every compact $n$-submanifold of $M$ (again possibly with boundary).

The following corollary is almost immediate if you believe either answer here (ignore my attempt at the non-compact case).

Corollary: Every orientable $3$-manifold is parallelizable.

Proof: For any compact 3-submanifold with boundary, the double is a compact orientable 3-manifold with empty boundary and hence has trivial tangent bundle. $\blacksquare$

The conjecture can be of course reformulated to a statement of the homotopy class of the classifying map from $M$ to the infinite $n$-Grassmanian. In fact as any smooth finite-dimensional real vector bundle can be embedded into $\Bbb R^M$ smoothly, one can consider homotopy classes of a classifying map from $M$ to some finite $n$-Grassmanian $Gr_n(\Bbb R^M$). This step fails for complex vector bundles, and there are examples of complex line bundles over (a 3-manifold even!) where the analogous complex conjecture fails (see here).

As the restriction of the classifiying map is the same as the classifying map of the restriction, and any finite simplicial(or CW) subcomplex of $M$ is inside some compact $n$-submanifold of $M$, we have that the classifying map is null-homotopic when restricted to any finite simplicial(or CW) complex where $M$ is thought of as a simplicial(or CW) complex. Now the relation to phantom maps becomes quite clear. The hope now is that I have said enough about this classifying map to be sure it is not a phantom, and there are purely homotopy theory reasons why.

Question: If $f:M\rightarrow G$ is a map of simplicial (or CW) complexes with $f$ restricted to every finite subcomplex of $M$ is null-homotopic with $M$ finite dimensional and $G$ finite, is $f$ necessarily null-homotopic?

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  • $\begingroup$ I may have been a little loose with my use of simplicial. Manolescu proved that not every high dimensional manifold admits such a structure. If you want to assume $M$ is PL, that is fine by me, though I just realized I do not know how to show $Gr_n$ is... $\endgroup$ Feb 13, 2015 at 4:44
  • $\begingroup$ Every smooth manifold carries a PL structure. $\endgroup$
    – user98602
    Feb 14, 2015 at 19:47

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