Assume that $P\in \mathbb{C}[z]$ is a polynomial of degree $n$ with $n$ distinct roots $z\_1,z\_2,\ldots,z\_n$.


We identify $\mathbb{R}^3$ with $\mathbb{C}\times \mathbb{R}$. Put $a\_i=(z\_i,0)$.


Then $\mathbb{R}^3 \setminus \{a\_1,a\_2,\ldots,a\_n\}$ is the union of $\mathbb{R}^{3\geq0} \setminus \{a\_1,a\_2,\ldots,a\_n\} $ and $\mathbb{R}^{3\leq0} \setminus \{a\_1,a\_2,\ldots,a\_n\} $ where $\mathbb{R}^{3\geq0}=\mathbb{C} \times [0, +\infty)$ and $\mathbb{R}^{3\leq0}=\mathbb{C} \times (-\infty, 0]$.


This enable us to define a complex line bundle on $\mathbb{R}^3 \setminus \{a\_1,a\_2,\ldots,a\_n\}$ using clutching function $1/P(z)$.


Since the wedge sum of $n$, $S^2$ is a deformation retract of $\mathbb{R}^3 \setminus \{a\_1,a\_2,\ldots,a\_n\}$, we get a line bundle on the wedge sum of the spheres. 



> 
> Does the isometric class of this line bundle depend on choosing initial polynomial $P(z)$? What is the explicit formulation of the (first) Chern class of this line bundle, in $\oplus\_{i=1}^n \mathbb{Z}$, the cohomology of the base space?( In terms of the coefficients of $P(z)$)
> 
> 
> 



A line bundle on the wedge sum of spheres associated to a polynomial $P(z)\in \mathbb{C}[z]$
OK, here's how I think it goes. No need for the residue theorem or anything like that.


Let $S\_k$ denote the sphere corresponding to $z\_k$. To compute the Chern class of the restriction of your line bundle to $S\_k$, it suffices to compute the degree of the map 
$$
f:S^1\to S^1,\qquad f(z) = \frac{|P(z)|}{P(z)} = \frac{\overline{P(z)}}{|P(z)|}
$$
obtained by restricting your clutching function $1/P(z)$ to a small circle $S^1\subset\mathbb{C}$ which encloses $z\_k$ and contains no other roots in its interior, and normalizing. But since this $f$ is the composition of the map $z\mapsto P(z)/|P(z)|$ of degree $n$ and the map $z\mapsto \bar{z}$ of degree $-1$, and since the degree of a composition is the product of the degrees, the degree of $f$ is $-n$. 


Therefore I think that the Chern class of your bundle over $\vee\_n S^2$ is $-n$ times the generator of each sphere. 


