Triangulation of fiber bundles

Question

Let us suppose we are given a fiber bundle $(E,B,F,p)$ where all spaces involved are triangulable and compact. Assume we choose a triangulation for the base B. I believe it is possible to give E a simplicial complex structure in such a way that both the map $p \colon B \to B$ and the transition maps are simplicial. So we would have a notion of "simplicial fiber bundle". However, i would like to cite this result and I do not find it in the literature.

Could you give me a reference?

I want to work with fiber bundles in the category of finite simplicial complexes and simplicial maps between them.

Thanks in advance

Answer 1

Corollary 2.2 of "Triangulation of fibre bundles" by H. Putz states:

• if $\pi\colon E\to B$ is a differentiable fiber bundle (each of $E$ and $B$ are differentiable manifolds), then there exist simplicial complex triangulations of $E$ and of $B$ so that $\pi$ becomes a simplicial map.