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The Stiefel-Whitney numbers are classical topological manifold invariants obtained by integrating some local quantity (a cup product of Stiefel-Whitney classes) over the manifold. Which Stiefel-Whitney numbers can be extended to classical invariants of topological manifolds with boundary?

Some background:

I don't know much about continuum topology, but I know the local formulas for $n-i$-cycles representing $\omega_i$ in an $n$-dimensional triangulation in [1] and their relation to framings restricted to the 1-skeleton, as well as the local formulas for cup products [2] and their intuitive interpretation as intersections. Combining both, we get a formula which computes the Stiefel-Whitney numbers locally from triangulations (i.e., they are a sum of a number at each vertex depending only on the link of that vertex).

I'm wondering whether (and which of) those local formulas can be extended to triangulations with boundary.

The Stiefel-Whitney classes are defined not only for manifolds but also for manifolds with boundary. $\omega_i$ restricted to the boundary of a manifold is simply $\omega_i$ of the boundary itself. However, the cup product in my naive intersection-form interpretation is not well-defined at the boundary since there is an ambiguity about whether an intersection of two cocycles on the boundary is considered inside or outside of the manifold.

The Stiefel-Whitney number consisting of only $\omega_n$ (in $n$ dimensions) is extendible to manifolds with boundary as it does not require a cup product (and is just the mod $2$ Euler characteristic). I also found a local formula a topological boundary of $\omega_1^2$ in $n=2$. Essentially, we can always define $\omega_i\cup X$ on the boundary if we have a trivialization of $\omega_i$ inside the boundary, i.e., a higher analogue of an orientation or a Pin structure. In the $\omega_1^2$ case, the local formula seems to be independent of the chosen trivialization (an orientation of the 1-dimensional boundary in this case).

[1] https://www.ams.org/journals/proc/1976-058-01/S0002-9939-1976-0415643-5/S0002-9939-1976-0415643-5.pdf

[2] https://www.maths.ed.ac.uk/~v1ranick/papers/steen5.pdf

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