It is easy to see that any (locally finite) graph is homotopy-equivalent to a trivalent graph. Moreover, this can be achieved by a local construction - take neighborhoods of vertices of degree $> 3$, remove them and glue in trivalent trees with needed number of outgoing edges.

I wonder whether the same$^1$ could be done for complexes of higher dimension (say, locally finite, or even finite). I seem to be able to construct finite list of neighborhoods able to approximate any $2$-dimensional complex (the proof is by induction on the gluing of $2$-dimensional discs).

$^1$ - I would like to have any complex to have the form Z/S, where Z is the complex of our fixed type and quotient morphism has collapsible fibers, so, local approximation in the simple homotopy type. I do not know whether it is needed but I would prohibit to increase the dimension, also.

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I didn't manage to find finite list of triangulations, though (I expect it is possible by analogy with the surfaces case where every triangulation admits local surgery that forces every link to have $\leq 7$ triangles).

I wonder whether this question was considered, and what is known in higher dimensions - possibly some kind of finiteness result?


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