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  1. Given a simplicial complex with only triangulation and only branching structure, is it enough to define Stiefel–Whitney class?

(Please provide Yes or No answers, and reasonings.)

  1. Given a fixed triangulation, with different branching structures, would the Stiefel–Whitney class depend on the choice of branching structure?

(Please provide Yes or No answers, and reasonings.)

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.

  • The simplicial complex may not need to be a manifold.

Do we have the same situations for Stiefel–Whitney class? Pontryagin class? How about other characteristic classes?

p.s. According to Tom Goodwillie's comment: "$𝑤_𝑗(𝑇𝑀)$ does not require a triangulation (a 𝑃𝐿 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; $𝑤_𝑗$ can be defined as the class that corresponds via the Thom isomorphism to $𝑆𝑞^𝑗$ 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 (non necessarily a manifold)?

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    $\begingroup$ Cross-posted $\endgroup$
    – Tim Campion ♦
    Dec 9, 2021 at 18:12
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    $\begingroup$ Is there some reference where I can read about branching structures? I'm confused by your description, because so far as I can see there is no reason for the link of a vertex to be a manifold, and so I don't know what it means to choose an orientation of the link. $\endgroup$
    – Tim Campion ♦
    Dec 9, 2021 at 18:14
  • $\begingroup$ One ref I know is Alexander_A_Gaifullin_2005_Russ._Math._Surv._60_615_Computation of characteristic classes of a manifold from a triangulation of it maths.ed.ac.uk/~v1ranick/papers/gaifullin.pdf, but there also is a manifold, with some triangulations $\endgroup$
    – wonderich
    Dec 9, 2021 at 18:25
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    $\begingroup$ Usually, characteristic classes are invariants of (vector or principal) bundles, not manifolds, e.g. Stiefel-Whitney classes of a manifold depend on its tangent bundle. One can also define the Stiefel-Whitney classes in the topological case for the tangent microbundle, see e.g. mathoverflow.net/questions/243629/…. What bundle are you considering over your simplicial complex? What is a wish list (a set of properties) for the invariant you are trying to define? $\endgroup$
    – Igor Belegradek
    Dec 9, 2021 at 23:05

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