Euler Class is JUST the 1st Obstruction to the Existence of a Non-vanishing section.

/ Alexander

It is well-known from algebraic topology and differential geometry that the Euler class e(E) of a vector bundle E\longrightarrow M is the obstruction of the existence of a no-where vanishing section s(x).

It has lots of formalisms, for example, we can define it via an obstruction-theorotical construction. To apply the obstruction theory, we can view a nowhere vanishing section s(x) as a section of the Stiefel manifold bundle V_1(E) associated to E , where each fiber was simply replaced by the Steifel manifold V_1(E_x). By a homotopical argument, we know that no matter when E is a complex or real vector bundle, V_1(E_x) is homotopic to S^{n-1}, where n=\mathrm{rank}\,E, and hence

\pi_{n-1}(V_1(E_x))\cong \pi_{n-1}(S^{n-1})\cong\mathbb{Z}

Hence for each (nowhere vanishing) section s defined on a 0-skeleton of M can be extended to the (n-1)-skeleton, but it may fail to be extended to the n-skeleton, since if s can be extended to some n-cell \sigma, one must have s(\partial \sigma) is trivial in \pi_{n-1}(V_1(E_x)). However, this is not always the case since the later homotopy group is nontrivial. The Euler class e(E) is defined to be the obstruction of this extension:

\begin{aligned}e(E): C_{n}(M;\mathbb{Z})& \longrightarrow \pi_{n-1}(V_1(E_x))\cong\mathbb{Z}\\ \sigma &\mapsto [s(\partial\sigma)]\end{aligned}

It can be shown that e(E) is well-defined, that is independed of the choice of the section s, and which is in H^n(M;\mathbb{Z}).

Another interesting construction (see here) is by applying Poincaré duality, but we need to assume that M is oriented and compact. Let s: M\longrightarrow E be a section such that it intersects transversally with M: s\pitchfork M so that the zero locus of s is indeed a submanifold and presenting a homology class [Z] in H_{m-n}(M;\mathbb{Z}), where \dim M=m. We take the Poincaré dual \mathrm{PD}([Z])\in H^{n}(M;\mathbb{Z}) to be the Euler class of E. It can be shown that it is well-defined and coincides with the first one.

It is also worth to mention that when E is a compelx vector bundle, the Euler class coincides with the top Chern class c_n(E) of E, and if E is real, the Euler class modulo 2 reduction coincides with its top Stiefel-Whitney class w_n(E).

However, the Euler class is just the 1st obstruction, not the total obstruction. Which is to say, there exists a vector bundle with vanishing Euler class but does not admit a nowhere vanishing section. Here is a counter-example.

Let’s consider a rank 3 real vector bundle E over S^4 with \det E=\bigwedge\nolimits^3 E trivial, one must have e(E)=0 since H^3(S^4;\mathbb{Z}) is trivial, but this E admits a section with no zeros if and only if E is trivial.

In fact, we assume s is such a section, then s spans a trivial line bundle L so that E splits as E'\oplus L, where E' is a rank 2 real vector bundle. Since

\det E=\det E'\otimes L

is trivial, we deduce that \det E' is also trivial, hence E' has the structure group SO(2)\cong S^1. However, the S^1 bundles over S^4 were classified by \pi_3(S^1)\cong 0, which implies E must be trivial.

In fact, there does exist a non-trivial rank 3 vector bundle on S^4 with vanishing 1st Stiefel-Whitney class, since \pi_3(SO(3))\cong\pi_3(\mathbb{RP}^3)\cong \mathbb{Z} which is non-trivial.

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