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
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.