3

From my very limited understanding, cohomology sometimes appears in robotics and sensing applications. For example, one might consider the de Rham cohomology of the configuration space. Often, one simply just checks that it is trivial so we can use naive gradient descent and call it a day. In sensing, one forms the Čech cohomology from the nerve complex of "charts" of the sensor network. If certain groups are trivial, then that implies no lack of coverage.

However, in both of these cases the application is quite rudimentary. Unlike, say, using specific Stiefel-Whitney classes or the cup product structure to discern spaces in topology, or detecting possible interesting anomalies/phenomena in electromagnetism, these two applications do not obtain information from the structure or classes themselves, only from whether they are trivial or not. Obviously, there are probably real-world reasons to come up with a simple "Yes/No" checklist, and the latter example can be modified to give the number of coverage holes and islands, but I am wondering if anybody could share examples where cohomology is used in a more nuanced way in robotics and sensing.

Adam Wang
  • 347
  • Have you had a look at the course notes Configuration spaces, braids, and robotics by Ghrist yet? Perhaps the techniques considered there are too "rudimentary" for what you're looking for though. – pyridoxal_trigeminus Aug 09 '25 at 22:48
  • These papers (1 related to topological complexity) and (2 relating to TDA) use SW classes and are also potentially in the ballpark of what you're looking for. – pyridoxal_trigeminus Aug 09 '25 at 22:56
  • @R.H.Vellstra Thanks for linking those. I would consider the course notes to be more in the realm of representation theory, but it is algebraic topology nevertheless — and not rudimentary at all. The second TDA paper is quite interesting, it sounds like an algorithm to extract information about the cup product from point clouds which I did not know existed! – Adam Wang Aug 10 '25 at 01:54
  • @R.H.Vellstra The first paper is exactly what I was looking for though. So it sounds like we may view all possibilities with $n$ obstructions in some system as a configuration space $B$. To each point in $B$ the swarm should have some response, which we formalize as a configuration bundle $X\to E\to B$. This is then when I don't fully understand. The paper then defines two different notions of topological complexity, which from some preliminary reading of other sources sounds like a "lower-bound" on the – Adam Wang Aug 10 '25 at 01:54
  • @R.H.Vellstra minimal number of (smooth?) if-then statements to cover everything. Then cup products are some how shown to lower bound this quantity again. I am not sure if this understanding is correct. If you have the time to write up an answer explaining this I would be very grateful and gladly accept it. – Adam Wang Aug 10 '25 at 01:55
  • Somebody also voted to close this question with the stated reason "Missing context or other details"? Here is a very similar question with much less context and details and that question was not closed — https://math.stackexchange.com/questions/235524/what-are-the-applications-of-differential-geometry-in-robotics – Adam Wang Aug 10 '25 at 02:10

0 Answers0