The Euler characteristic is the unique locally determined numerical homotopy invariant of finite complexes (Q1186082)

This paper shows that, up to a multiplicative constant the Euler Characteristic is the unique numerical homotopy invariant of a finite simplicial complex that has a local formula. The proof is based on the observation that any such invariant \(\rho\) must satisfy the additive formula. \(\rho(K_ 0\cup K_ 1)=\rho(K_ 0)+\rho(K_ 1)-\rho(K_ 0\cap K_ 1)\). The author observes that if the complexes are restricted to be closed manifolds the result no longer holds.