Given a simplicial complex with only triangulation and branching structure, is it enough to define Stiefel–Whitney class and Pontryagin class?
If so, could you explain how to obtain these characteristic classes via a simplicial complex with only triangulation and branching structure?
A triangulation is given by the simplicial complex.
A branching structure is a choice of the orientation of each link in the simplicial complex, so that there is no oriented loop on any 2-simplex.
p.s. According to Tom Goodwillie: "$_()$ does not require a triangulation (a structure). The tangent microbundle of a topological manifold has Stiefel-Whitney classes. In fact, even a (stable) spherical fibration on a space X has SW classes; $_$ can be defined as the class that corresponds via the Thom isomorphism to $^$ of the Thom class." If this is true, what do Stiefel–Whitney class and Pontryagin class require in order to be defined on a simplicial complex?