A Brouwer fixed-point theorem for graph endomorphisms

@article{Knill2012ABF,
  title={A Brouwer fixed-point theorem for graph endomorphisms},
  author={O. Knill},
  journal={Fixed Point Theory and Applications},
  year={2012},
  volume={2013},
  pages={1-24}
}
  • O. Knill
  • Published 2012
  • Mathematics, Computer Science
  • Fixed Point Theory and Applications
  • We prove a Lefschetz formula L(T)=∑x∈FiT(x) for graph endomorphisms T:G→G, where G is a general finite simple graph and ℱ is the set of simplices fixed by T. The degree iT(x) of T at the simplex x is defined as (−1)dim(x)sign(T|x), a graded sign of the permutation of T restricted to the simplex. The Lefschetz number L(T) is defined similarly as in the continuum as L(T)=∑k(−1)ktr(Tk), where Tk is the map induced on the k th cohomology group Hk(G) of G. The theorem can be seen as a generalization… CONTINUE READING
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