In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on fixed points of self-maps is treated in the adjacent section 15.3 (see theorem 5.3.2 on page 131).

Even though the sections are adjacent, no mention is made of the relation between the two theorems, perhaps because this is "obvious" (take the flow of the vector field for small time, then apply Lefschetz, etc).

Is there a framework that would allow one to deduce both results as a consequence of a more general theorem?

  • $\begingroup$ I'm not really sure what you're looking for here, since as you say (and abx supplies the details of), one is very quickly proved from the other. So the Lefschetz theorem is the "more general theorem". $\endgroup$ – Allen Knutson Jan 1 '14 at 14:06
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    $\begingroup$ @AllenKnutson, as you may have noticed the details of the argument involve choosing local coordinates, solving a differential equation for "small" time, but in the end one wants to count what is a global invariant. While all this is elementary to someone who is already familiar with these techniques, one might wonder if a more conceptual connection exists between these results via a more general one that would directly specialize to both. $\endgroup$ – Mikhail Katz Jan 1 '14 at 14:44
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    $\begingroup$ How about: Lefschetz is about intersecting $M_\Delta \cap graph(f) \subseteq M\times M$, whereas Poincare-Hopf is about intersecting the zero section with $\sigma^{-1}(0)$ inside $TM$, where $\sigma:M\to TM$ is a section of the tangent bundle. Then one can repackage the analysis: instead of the small-time flow of the vector field, identify a tubular neighborhood of $M_\Delta$ with $TM$. Now both theorems are about intersecting $M_\Delta$ with another submanifold of $M\times M$. $\endgroup$ – Allen Knutson Jan 2 '14 at 1:05
  • $\begingroup$ Thanks. I was thinking that for more general spaces $M$ one might still have the Lefschetz result (provided $M$ is at least a Poincare duality space, or alternatively is a sufficiently "nice" algebraic variety), but defining tangent bundles is a problem. In other words, the implication Lefschetz theorem $\rightarrow$ Poincare-Hopf theorem seems to depend on possibly extraneous smooth structure. Perhaps one could have a "coarser" notion of a vector field that would allow both theorems to be stated in a single framework. $\endgroup$ – Mikhail Katz Jan 2 '14 at 9:01
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    $\begingroup$ Just a note: It isn't entirely obvious that, for a compact manifold $M$ and a vector field $V$, there is a time $t$ such that the only fixed points of flow along $V$ at time $t$ are the zeroes of $V$. This was answered in math.stackexchange.com/questions/48074 $\endgroup$ – David E Speyer Jan 2 '14 at 15:30

The Poincaré-Hopf theorem is a consequence of the Lefschetz fixed point theorem if you accept the fairly standard fact that the Euler characteristic of a compact manifold $M$ is equal to the self-intersection of the diagonal $\Delta \subset M\times M$. Here is why :

Since $M$ is compact, the vector field $X$ gives rise to a flow $(\varphi _t)_{t\in\mathbb{R}}$; the fixed points of $\varphi _t$ are the zeros of $X$. Suppose these zeros are isolated, and let $p$ be one of them. In local coordinates, we have $\varphi _t(x)= x+tX(x)+o(t)$ around $p$, hence $d\varphi _t(p)=I+t\,dX(p)+o(t)$. This implies that for small $t$, the index of $X$ at $p$ is equal to the index of $p$ as a fixed point of $\varphi _t$. Lefschetz theorem tells you that the sum of these indices is the intersection number $(\Delta .\Delta _t)$, where $\Delta _t$ is the graph of $\varphi _t$. This integer depends continuously on $t$, hence is constant, hence equal to $(\Delta .\Delta) =e(M)$.


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