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It is known that some topological manifolds, even compact and simply-connected ones, do not have admit a triangulation. One example is the E8 manifold in a dimension as low as $4$.

I am trying to understand why a topological manifold without triangulation would ever be possible, since this property appears counter-intuitive.

A topological $n$-manifold looks locally like $n$-dimensional space, so it seems intuitive that some (finite) local triangulations can be merged (after subdivision) to produce a simplicial complex on the whole of the manifold.

What particular feature prevents a compact manifold from having a triangulation? Is there an simplified geometric description of this counter-intuitive feature?

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    $\begingroup$ If you think about "merging" more, you will realize that it is easier said than done. Even triangulating 1-dimensional manifolds takes some work. Triangulating surfaces (at least using standard arguments) requires some nontrivial tools limited to dimension 2, etc. You may want to think also about existence of a smooth structure and the idea of "merging" two atlases. Then look at Milnor's examples where the obstruction to "merging" lies in the fact that not all (orientation-preserving) diffeomorphisms of $S^6$ are isotopic to the identity. $\endgroup$
    – Moishe Kohan
    Commented Apr 29 at 8:15
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    $\begingroup$ How do you merge? Two embeddings of triangles (coming from different local triangles) can intersect in a nasty, fractal like manner. Think topologist sine curve with the X axis. $\endgroup$
    – Thomas Rot
    Commented Apr 29 at 9:37
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    $\begingroup$ Every smooth manifold admits a triangulation. So non-triangulable manifolds are inherently somewhat pathological and counter-intuitive. I assume you know that the Kirby–Siebenmann class is an obstruction for topological manifolds to have a PL-structure. Are you looking for something more than that? $\endgroup$
    – Timothy Chow
    Commented Apr 30 at 1:38

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