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I am hoping someone will be willing to help me take a look at if this conterexample works?

Let $X$ be an oriented compact manifold and $f : X \to X$ a map. Suppose $W$ is a compact oriented manifold with boundary $\partial W = X$ and $F : W \to W$ a map whose restriction to the boundary is $f$. Then the global Lefschetz number of $f$ vanishes.

Counterexample:

Consider $W = D$, the unit disk. And $X = \partial W = S^1$. $$f = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.$$

Then the only fixed point is the origin, and clearly $df_x - I = f(x) - I$ would not vanish $\forall \theta \neq 2k\pi, k \in \mathbb{Z}$.

1LiterTears
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    I think you got confused between $f$ and $F$. The theorem says that global Lefschetz number of $f$ vanishes. In your counter example rotation is $F$ whose restriction to $S^1$ which is $f$ in the theorem does not even have any fixed point. – tessellation Aug 20 '13 at 18:36

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