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Let $K$ be a simplicial complex and let $f:K \to K$ be a simplicial map.

Assume the existence of an embedding $\iota: K \hookrightarrow \mathbb{R}^n$ of this complex into Euclidean space and note that $f$ induces a continuous self-map of $\iota(K) \subset \mathbb{R}^n$ which I will call $g$.

The Lefschetz number of $g$ is known by the trace formula applied to the endomorphisms of homology groups $H_*(K)$ induced by $f$. But can we also recover local fixed point information? That is,

can we determine the fixed point indices, i.e., the topological degree of $x - g(x)$ at each fixed point of $g$ only using knowledge of $f$? If so, how?

If this is a standard result, I apologize. I couldn't find the relevant theorem in Granas and Dugundji's encyclopedia... any help is appreciated.

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  • $\begingroup$ Is this not the local degree which is calculated in Allen Hatcher's topology textbook (chapter 2, pg136)? $\endgroup$
    – Chris Gerig
    Commented Jun 28, 2012 at 1:06
  • $\begingroup$ Maybe, but this is not clear to me. That entire section considers maps from the sphere $S^n$ to itself. How does one apply this to the situation above unless $f$ maps a given simplex to itself? $\endgroup$
    – user24459
    Commented Jun 28, 2012 at 1:44

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