# Euler Characteristic of fiber bundle using Differential Geometry

According to the last answer to this question. It seems there is a way, using Differential Geometry, to proof that Euler Characteristic is multiplicative on fiber bundles i.e.: $$\chi(E)=\chi(F)\cdot \chi(B)$$ where $p\colon E\to B$ is a smooth fiber bundle with fiber $F$.

My question: Could you give me a reference (article or book) for a proof of the multiplicativity of Euler characteristic on smooth fiber bundles which uses Riemanninan Geometry or Differential Geometry.

Remark: I am aware of the proof using spectral sequences and the combinatorial one.

• Trying to solve your question, I ended up searching for relations between holomogy/cohomology groups in the fiber bundle context since $$\chi(X) =\sum_{i=0}^\infty (-1)^i H_i(X).$$ Do you know anything in this area? I am aware of the injection between $H_i(E)$ and $H_i(B)$, but this does not seem to help. Jun 29, 2017 at 19:30
• As a possible route, have you thought about how you'd use the Poincaré-Hopf theorem to prove this in the case of a trivial bundle (i.e., product of two manifolds)? Jun 29, 2017 at 21:03
• @AlexisLeroux-Lapierre this paper by Spanier along with the Kunneth Formula is enough to obtain the result using (co)homology and betti numbers but OP wants Dif Geo proof Jun 30, 2017 at 1:34
• @TedShifrin It seems that the Poincaré-Hopf theorem could be used in the non-trivial case too, don't you think? Jun 30, 2017 at 1:40
• Both @AlexisLeroux-Lapierre and @ user253929 thanks for pointing out that argument. I was not aware of it. Jun 30, 2017 at 12:20

If you know about Morse Bott functions, the proof is simple. If a $f:B\to \bf R$, is a Morse function, then $F=f\circ \pi : E\to \bf R$ is a Morse-Bott function whose critical submanifold are exactly the fibers over critical point and the same indices. Thus $\chi (E)= \sum _c ind(c) \chi (F_c)= \chi (F)\times \chi (B)$.
Here is a self-contained proof using Poincare-Hopf theorem (as suggested by Ted). Let $\nabla$ be a connection on the bundle $p: E\to B$. Take a nondegenerate vector field $X$ on $B$ and lift it to a vector field $Y$ on $E$ using $\nabla$. Let $b_1,...,b_n$ denote the singular points of $X$; set $F_i:= p^{-1}(b_i), i=1,...,n$. Take small (pairwise disjoint) tubular neighborhoods $p^{-1}(U_i)$ of the fibers $F_i$, $i=1,...,n$. These neighborhoods admit a product structure $F_i\times U_i$ consistent with the fibration $p$. Pick a nondegenerate vector field $Z$ on $F\cong F_i, i=1,...,n$, extend it to each $U_i$ (using the product decomposition) and then multiply by pull-backs (via $p$) of suitable bump-functions, supported on $U_i$'s. Then extend the resulting vector field $W$ to the rest of $E$. Lastly, take the vector field $V=Y+W$. The set $Sing(V)$ of singular points of $V$ is the disjoint union of copies of $Sing(Z)$ in $F_1,...,F_n$. At each point $q\in F_i\cap Sing(V)$, $$index(V,q)= index(Z,q) \times index(X, b_i)$$ (this follows from the fact that the determinant of a square block-diagonal matrix with blocks $A_1, A_2$ equals $det(A_1)\times det(A_2)$).
Now, what's left is just to count the number of singular points of $V$ (with sign): Each fiber $F_i$ contributes $$index(X,b_i)\times index(Z)= index(X,b_i)\times \chi(F).$$ Summing up over all points $b_i$ we obtain $$index(V)= index(X)\times \chi(F)= \chi(B)\times \chi(F).$$ qed