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In Bott and Tu's Differential Forms in Algebraic Topology, the Lefschetz number of a map $f:M\to M$ between an oriented compact manifold $M^m$ is defined, as in any algebraic topology text, to be $L(f)=\sum_q (-1)^q tr(f^*|_{H^q(M)})$, where we are using de Rham cohomology groups here. In its Exercise 11.26 (p.129), Bott and Tu showed that $$ \int_\Delta \eta_{\Gamma_f}=L(f) $$ where $\Gamma_f$ is the graph of $f$ in $M\times M$, $\Delta$ is the diagonal of $M$ in $M\times M$, and $\eta_S\in H^{n-s}(N)$ is the Poincare dual of a closed submanifold $S^s\subset N^n$ characterized by the property $$ \int_S i_S^*\omega=\int_{N}\omega\wedge \eta_S $$ for all $\omega\in H^s_c(N)$. It is already shown in this post that if $S^s,L^\ell\subset N^n$ closed submanifolds such that $s+\ell=n$ and $S\pitchfork L$, then the intersection number $I(S,L)$ in differential topology (this is defined in Gullemin and Pollack's differential topology text) satisfies $$ I(S,L)=\int_N \eta_S\wedge \eta_L. $$

Here is my question: If we take the result in Bott and Tu, by the property of Poincare dual and the intersection number result, we have $$ L(f)=\int_\Delta \eta_{\Gamma_f}=\int_{M\times M}\eta_{\Gamma_f}\wedge \eta_\Delta=I(\Gamma_f,\Delta). $$ However, I looked up on Gullemin and Pollack. They defined the Lefschetz number $L(f)$ to be $$L(f)=I(\Delta,\Gamma_f).$$ So it would seem that two definitions of the Lefschetz number differ by a sign $(-1)^m$, where $m$ is the dimension of $M$. However, I must be doing something wrong, otherwise I cannot prove part (c) of Exercise 11.26 of Bott and Tu. I can show that if $L(f)=I(\Delta,\Gamma_f)$, then we have $$ L(f)=\sum_{p\in\text{Fix}(f)}sgn(\det(df_p-I)) $$ but not if $L(f)=I(\Gamma_f,\Delta)$ which is suggested by Bott and Tu, since I would get an extra sign $(-1)^m$. Can anyone point out what I am missing here? Any help would be appreciated.

Tianyi Wang
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  • It's probably just two (pairs of) authors using different conventions. The key property of the Lefschetz number only depends on whether or not it vanishes, which doesn't care about the sign. It's also possible (based on what happened when you did the exercise) that some authorial sloppiness was involved. – Tabes Bridges Jan 15 '24 at 20:45
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    Well there is an important identity which should determine the sign: when $f$ is the identity map then $L(f)$ should equal the Euler charateristic. – Lee Mosher Jan 15 '24 at 20:52
  • @TabesBridges Thank you for your reply. But somehow this doesn't feel right because, for example, the index of a vector field at one of its zero is defined to be the local Lefschetz number of the derivative of the flow, so in this version, the index of a vector field would always be off by some sign, which seems suspicious... – Tianyi Wang Jan 16 '24 at 01:39
  • @LeeMosher Also when $m$ is odd, then in both definition $I(\Delta,\Delta)=0$, and so is the Euler characteristic $\chi(M)$. When $m$ is even, the factor $(-1)^m$ does not matter, so not sure how this would help. – Tianyi Wang Jan 16 '24 at 03:42
  • Hmmm, that's a very good point. I withdraw my comment. – Lee Mosher Jan 16 '24 at 14:20
  • I think there's a sign problem with the statement $I(S,L) = \int_N \eta_S\wedge\eta_L$. Try $N=S^1\times S^1$ and let $S=S^1\times{q_0}$, $L={p_0}\times S^1$. – Ted Shifrin Jan 18 '24 at 19:27
  • I think the signs are messed up in Bott-Tu from the outset with the definition. We should have $\int_S \omega = \int_M \eta_S\wedge\omega$. Similarly, as we need to take the intersection number of $Z$ (the zero section) and $s$ (a generic section) to calculate the appropriate index of a zero, we need to take the intersection number of $\Delta$ and $\Gamma_f$. Signs are always something one has to check carefully — and, in complex geometry, whether we have a factor of $\sqrt{-1}$ or of $1/\sqrt{-1}$ :) – Ted Shifrin Jul 04 '24 at 17:21

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