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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!

StormyTeacup
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    I would start with the Euler-Poincare formula for a vector field. Does that have a geometric meaning to you? – Lee Mosher Feb 20 '21 at 03:02
  • Not as firm a meaning as I really would like it to have. Basically, my understanding of it is no more sophisticated than that it is the alternating sum of the number of $k$-cells. The standard vertices minus edges plus faces etc., etc. in other words. – StormyTeacup Feb 20 '21 at 03:07
  • @StormyTeacup Observe the case when the map $f:X\to X$ is the identity. I think you'll see the connection between Lefschetz number and Euler characteristics. – Kevin.S Feb 20 '21 at 03:38

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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.

Sam Freedman
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