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