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The two following theorems appear to be contradictory. Both are proved in Milnor's "Topology from the differentiable viewpoint." I'm sure I'm overlooking something incredibly trivial:

  1. Let $f$ be a smooth map from a smooth compact oriented manifold $K$ to a smooth connected oriented manifold $N$, both of dimension $m-1$. If $K$ is the oriented boundary of some smooth compact oriented $m$-manifold (with boundary) $Y$ and if $f$ extends to a smooth map from $Y$ to $N$ then the degree of $f$ (i.e. the sum of the signs of the determinants of the Jacobians at the elements of the preimage of any regular value) is zero.

  2. (Poincaré-Hopf) If $M$ is a smooth compact manifold and $V$ is a smooth vector field on $M$ with isolated singularities then the degree of the vector field is equal to the Euler characteristic of $M$.

Now consider the setting of the second statement and assume $M$ is oriented. delete a small open ball around each zero of $V$. This gives a manifold with boundary $Y$ and by normalizing $V$ we get a map $g$ from $Y$ to the $(m-1)$-sphere. Now if $K$ is the boundary of $M$ then the first statement says the degree of $g$ restricted to $K$ is zero while the second statement says it is the Euler characteristic of $M$.

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$K$ is the domain of $g$, there is no extension. The natural way to extend the definition of $g$ to all of $Y$ would be to trivialize the tangent bundle, and if you could do that, certainly the Euler characteristic would be zero. – Ryan Budney Aug 29 '11 at 3:20
thanks for your help ryan. maybe this doesn't fix the problem but let's assume M was imbedded in some Euclidean space. Now g makes sense on all of Y regardless of the whether the tangent bundle is trivial right? – Tim kinsella Aug 29 '11 at 3:33
$g$ does make sense, but not of something that you can take the degree of, since the domain and range have the wrong dimensions. – Ryan Budney Aug 29 '11 at 4:00
but I only speak of the degree of g restricted to K. K has dimension m-1 and the target space is the m-1 sphere. im clearly missing something basic and thank you for your patience. – Tim kinsella Aug 29 '11 at 4:06
maybe the cofusion is a result of my bad notation: the M from the first proposition is not the M from the counter example – Tim kinsella Aug 29 '11 at 4:07

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