The Lefschetz Number: A Geometrical Interpretation Well, my question is rather brief and straightforward: does anyone know of any good geometrical interpretation of the Lefschetz number, specifically, if $f$ is a continuous map $X \rightarrow X$, $X$ being a compact triangulable space, then the Lefschetz number is defined by
$$\Lambda_f : = \sum_{k \geq 0} (-1)^k \text{tr} (f_* : H_k(X) \rightarrow H_k(X)).$$
I just kind of like having these constructs be slightly more tangible in my mind than mere abstract formulae.
Look forward to your answers!
 A: If you allow me to assume that permutations are geometric enough, we can  give a geometric relationship between trace and fixed points. Let $\sigma \in S_n$ be a permutation, and let $P_\sigma$ be its corresponding permutation matrix. A 1 along the diagonal of $P_\sigma$ corresponds to an index $k$ such that $\sigma(k) = k$, i.e a fixed point of $\sigma$. We see that $\text{tr}(P_\sigma)$ is counting the fixed points of $\sigma$.

With this in mind, we can follow this sketch to show that
$$f\ \text{has no fixed points}\ \implies \Lambda_f = 0.$$
First, after possibly subdividing the triangulation of $X$, we can use simplicial approximation to argue that $f$ is homotopic to a simplicial map $g$ that fixes no simplices. By the above comments, the induced simplicial chain maps $g_* : C_k(X) \to C_k(X)$ all have trace $0$. It follows that
$$\sum_{j \ge 0} (-1)^j \text{tr}(g_*: C_k(X) \to C_k(X) ) = 0.$$
This looks like the Lefschetz number! In fact, we can use some homological algebra (which goes by the Hopf Trace formula here) to argue
$$\sum_{j \ge 0} (-1)^j \text{tr}(g_*: C_k(X) \to C_k(X) ) = \sum_{j \ge 0} (-1)^j \text{tr}(g_*: H_k(X) \to H_k(X) ) = \Lambda_g = \Lambda_f.$$
The last equality follows from the homotopy invariance of homology.
