Notation clarification in fibre bundles

Question

I am trying to comprehend the following material, I have some doubts mostly regarding the notations.

Chern Classes: Throughout this section, we will use $\mathbb{Z}$ coefficients and let $\xi=(E, X, \pi)$ denote a $d$ 'dimensional vector bundle with total space $E$, base space $X$ and projection $\pi$. We also use $E^{\#}$ to denote $E \backslash s_0(X)$, where $s_0: X \rightarrow E$ is the zero section. In the case where $\xi$ admits a $\mathbb{Z}$-orientation $\left\{\varepsilon_x\right\}_{x \in X}$, recall that there is a unique class $u_{\xi} \in H^d\left(E, E^{\#}\right)$ (the Thom class) which restricts to $\varepsilon_x$ on each fibre. The Euler class $e(\xi) \in H^d(X)$ is then defined by the image of $u_{\xi}$ under the composition $$ H^d\left(E, E^{\#}\right) \longrightarrow H^d(E) \stackrel{s_0^*}{\sim} H^d(X) . $$ This has the property that, if $\xi^{\prime}=\left(E^{\prime}, X^{\prime}, \pi^{\prime}\right)$ is another oriented $d$-dimensional real vector bundle and $(F, f): \xi \rightarrow \xi^{\prime}$ is an orientation-preserving bundle morphism, then $f^* e\left(\xi^{\prime}\right)=e(\xi)$. We say that $e$ satisfies the naturality condition.

Please let me know what are following

1. $\left\{\varepsilon_x\right\}_{x \in X}$, particularly what are the values $\varepsilon$ can take here
2. In here $H^d\left(E, E^{\#}\right)$ the cohomology $H^d$ takes two arguments, while in here $H^d(E) $ there is only one argument, so what are these spaces with regards to the cohomology.

In here, I assume the following correct, if not, please correct me.

1. $\mathbb{Z}$ coefficient means the coefficients are integers
2. $E \backslash s_0(X)$ means the set $E$ with the zero section removed
3. $s_0^*$ stands for the pull back of the zero section map $s_0$

Answer 1

1. Let $\pi : E \to X$ be a real rank $d$ vector bundle and let $E_x := \pi^{-1}(x)$ be the fiber of $E$ over $x$. A $\mathbb{Z}$-orientation on $E$ is a choice of generator $\varepsilon_x$ of $H^d(E_x, E_x\setminus \{s_0(x)\}; \mathbb{Z})$ for every $x \in X$ such that for each $x \in X$, there is an open set $U \subseteq X$ and a cohomology class $u \in H^d(\pi^{-1}(U), \pi^{-1}(U)\setminus s_0(U);\mathbb{Z})$ with the property that $u|_{E_x} = \varepsilon_x$ for all $x \in U$. A good reference for this material is Milnor and Stasheff's Characteristic Classes (for this specific question, chapter $9$).
2. Here $H^d(E, E^{\#})$ is a relative cohomology group. There is a map of pairs $(E, \emptyset) \to (E, E^{\#})$, so there is an induced map of relative cohomology groups $H^d(E, E^{\#}) \to H^d(E, \emptyset) = H^d(E)$.

The three statements at the end of your post are correct.