Lefschetz-Hopf fixed point theorem
According to Wikipedia, the Lefschetz-Hopf fixed point theorem states that if $X$ is a compact triangulable space and $f:X\rightarrow X$ is continuous and has a finite number of fixed points, then$$ \sum_{x\in X,f(x)=x}i(f,x)=\sum_{k\geqslant 0}(-1)^k{\rm tr}\left(f_*:H_k(X,\mathbb{Q})\rightarrow H_k(X,\mathbb{Q})\right) $$where $i(f,x)$ denotes the index of the fixed point $x$. I can't find out how $i(f,x)$ is defined. I am also looking for a proof of this theorem. Thanks in advance.
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$\begingroup$$i(f,x)$ is what is called the index, you may find the definition and several related things about index in the beautiful book on Differential Topology by V. Guillemin and A. Pollack.
Just to point, a variant of the above theorem is also true for compact simplicial complex for which you can find a proof in the Algebraic Topology book of A.Hatcher.
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