To expand upon Douglas Zare's comment a bit, in the book

> Farber, Michael. *Invitation to topological robotics*. European Mathematical Society, 2008,

Farber uses C&P (cut & paste) surgery and the Grothendiek ring $\mathfrak{C}$&$\mathfrak{P}$
to compute Euler characteristics of polyhedral configuration spaces (pp.55-56):
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![C&P][1]
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  [1]: https://i.sstatic.net/h0r6f.png