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If $f\colon X\to X$ is a self-map of a nice space with isolated fixed points, then the Lefschetz fixed point theorem relates a global number to local numbers. Some write: $L(f)=\sum_{x\in \mathrm{Fix}(f)}L_x(f)$.

It seems to me dangerous to use the letter $L$ for both terms. In the basic case, the symmetry is nice, but what if the two types mix? For example, take a map $p\colon E\to B$ compatible with self-maps $f_E$ and $f_B$. For each $b\in \textrm{Fix}(f_B)$, $f_E$ restricts to a self-map of $p^{-1}(b)$; call this $f_b$. Then $L(f_E)=\sum_{b\in\textrm{Fix}(f_B)}L_b(f_B)L(f_b)$, multiplying the two types of terms. Isn't that confusing?

Indeed, not everyone uses $L$ for both sides. Wikipedia uses $\Lambda$ for the global term and $i$ (for index) for the local term. Goresky and MacPherson use $L$ for the global term and $n$ (for number?) for the local term. In contrast, Hatcher uses $\tau$ (for trace) for the global term, leaving $L$ for the local term.

Is any convention winning? Which side deserves the $L$? What should go on the other side? Does anyone use $L$ for both sides, but distinguish them, perhaps $L/l$ or $L/\Lambda$?

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    $\begingroup$ I don't see the problem. One of them has a subscript and the other doesn't. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 18, 2014 at 8:06
  • $\begingroup$ The main question is whether one notation is winning, which doesn't depend on whether you see a difference between the choices, or even if you don't care about consistency across papers. If no convention is winning, then there are subsidiary questions about what is better. Maybe those were a mistake. $\endgroup$
    – Ben Wieland
    Commented Dec 18, 2014 at 16:42

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At least in the degree-theoretic world, the index notation appears to be dominant. When we write the Lefschetz-Hopf theorem

$$L(f) = \sum_{x \in \text{Fix}(f)} \text{stuff}_x(f),$$

the $\text{stuff}$ in question is the fixed-point index of $f$ at $x$ which has a nice degree-theoretic description, so for instance the text

A Granas and J Dugundji, Fixed point theory, Springer (2003)

writes the local contribution as $I(f,U)$ where $U$ is any open set isolating $x$. A similar approach is taken in Section 6.20 of K Deimling's Nonlinear functional analysis.

Finally, a caveat: note that there are two (fairly nontrivial, at least for me) corrections to the Goresky-MacPherson paper which you've mentioned.

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  • $\begingroup$ Is the word "stuff" supposed to indicate that while the local term is consistently not L, it is not consistently any other letter? Or perhaps that there is more to notation than the choice of letter? $\endgroup$
    – Ben Wieland
    Commented Dec 30, 2014 at 19:31
  • $\begingroup$ @BenWieland The word "stuff" is a placeholder to indicate that there is no consistent notation for the local term. I've even seen cases where people give a completely arbitrary name to the local endomorphisms, e.g., $A_i$ and then write the formula as an alternating sum of $\text{tr}(A_i)$. $\endgroup$
    – Vidit Nanda
    Commented Dec 30, 2014 at 20:03

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