Can Euler Class be defined by the Splitting Principle for Real Vector Bundles? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:54:46Z http://mathoverflow.net/feeds/question/79728 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79728/can-euler-class-be-defined-by-the-splitting-principle-for-real-vector-bundles Can Euler Class be defined by the Splitting Principle for Real Vector Bundles? Acky 2011-11-01T17:08:01Z 2011-11-01T17:50:20Z <p>Let <code>$M$</code> be a manifold and <code>$S$</code> its sphere bundle with fiber <code>$\mathbb{S}^n$</code>. As we know, the notion of the Euler class is raised from the problem of finding a global form on <code>$S$</code> which restricts on each fiber to a generator of <code>$H_{\textrm{dR}}^n(\mathbb{S}^n)$</code>. One overcomes two obstructions, namely the orientability of the sphere bundle and the Euler class <code>$e(S)$</code> to find such a form. Now let <code>$N$</code> and <code>$M$</code> be two manifolds and <code>$f:N\rightarrow M$</code> be smooth, <code>$E$</code> is a vector bundle over <code>$M$</code>. The following functorial property of <code>$e(E)$</code> is well-known:</p> <p>$e(f^\ast E)=f^\ast e(E)$</p> <p>On the other hand, the splitting principle for cmplex vector bundles is wedely-known for its power on computing characteristic classes. In the real case, although the splitting principle does not hold in full generality, a weak version of this type of theorems can still be obtained: (cf. Lecture Notes in Mathematics, 638)</p> <p>In fact, let <code>$E$</code> be an even dimensional oriented real vector bundle over <code>$M$</code>. Then there exist a manifold <code>$N$</code> and a map <code>$g:N\rightarrow M$</code> satisfying the following two conditions:</p> <p>(1) the homomorphism <code>$g^\ast: H_{\textrm{dR}}^\ast(M)\rightarrow H_{\textrm{dR}}^\ast(N)$</code> is injective;</p> <p>(2) <code>$g^\ast(E)$</code> is a direct sum of plane bundles.</p> <p>Therefore one may expect that the Euler class can be defined for an arbitrary vector bundle <code>$E$</code> by first introducing the Euler class for plane bundles, then extending it to <code>$E$</code> using the splitting principle stated above. In fact, suppose that <code>$e(P_i)$</code> has been introduced for every plane bundle <code>$P_i$</code>, and <code>$g^\ast(E)$</code> is a direct sum of <code>$P_i$</code>, then <code>$e\big(g^\ast(E)\big)$</code> is well defined, therefore <code>$e(E)$</code> may be defined by</p> <p>$e\big(g^\ast(E)\big)=g^\ast e(E)$</p> <p>However, one still needs to show the existence of <code>$e(E)$</code> and its independence of the choices of <code>$N$</code> and <code>$g$</code>. It doen’t seem easy.</p>