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Looking at homology Euler class, I stucked with some intuitive construction:

Concider a bundle $E\rightarrow B$ of rank $k$, where the base space is a smoooth $n$-dimensional manifold. Take the zero section $\xi$. We want to concider something like intersection-product $\xi\cdot\xi$. So we take another section, say $s$ which is 'small perturbation' of $\xi$, namely they intersect on possibly small area. From construction $s^{-1}(0)\subset \text{im}(\xi)$. Now there is a confusion part. If our bundle is orientable (we can guarantee this via taking coefficients in $\mathbb{Z}_2$) "the zero locus $s^{-1}(0)$ has the structure of a cycle - part of this involves assigning multiplicities to the components in a certain way". I don't understand this sentence. What cycles do we concider? How the look like?

Thanks in advance

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The cycle is $s^{-1}(0) \cap \xi$ inside $\xi$. The point is that the Euler class should be defined with coefficient in $\Bbb Z/2 \Bbb Z$ because $E$ might be not orientable, so the intersection number (or say the multiplicity of the corresponding cycle) is not well defined as an integer, but only as an integer mod 2, because if it's not oriented you could deform a positive intersection into a negative one. On the other hand, complex manifolds are always naturally oriented so this is why the complex analogue, the Chern class, is defined with integer coefficients.

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  • $\begingroup$ By $\cap$ you mean usual intersection? What is the intersection number? $\endgroup$
    – Yelon
    Apr 24, 2018 at 22:55
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    $\begingroup$ @Yelon : I meant the usual intersection, because by hypothesis $s$ is a small perturbation of $\xi$ so it is transverse to $\xi$, and for two transverse cycles the intersection product coincide with the set-theoretical intersection. I just meant the multiplicity (this is the coefficient they are talking about). I was thinking to curves, when in this case we can look the intersection cycle as a number, but in general we get a cycle with possible multiplicity. $\endgroup$
    – Nicolas Hemelsoet
    Apr 24, 2018 at 23:17
  • $\begingroup$ Ok, I see more or less what is going on. And without creating another topic I'd ask one more thing. Let a base space $B$ as orientable, smooth $n$-dim manifold and concider $TB\rightarrow B$. Why then $e_H(TB)$ is multiplicity of a class of point $[*]$? I did an example with Mobius bundle, but cannot see how it works for arbitrary $B$ satisfied above assumptions? Or in another way: why it should be clear that $e_H(TB)\in H_0(B)$? $\endgroup$
    – Yelon
    Apr 25, 2018 at 14:33

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