Three fold covering of $\Bbb RP^2\lor \Bbb S^1$. I am trying to find $3$-fold covering of $\Bbb RP^2\lor\Bbb S^1$. So what I guess is the following:---
Choose three distinct points $a,b,c$ on a cricle, then the three open arcs $ab,bc,ca$ is our three one cells and the points $a,b,c$ are three $0$-cells. Now wedge three copies of $\Bbb RP^2$ at the points $a,b,c$. And this will give us the required $3$-fold covering. Am I right?
My next question is, can I find a $3$-fold covering by performing different operations like wedge, just on the spaces $\Bbb S^2$ and $\Bbb S^1$. Any help will be appreciated.
 A: If you have coverings $p_i : X_i \to Y_i$, then you get a covering $p_1 \times p_2 : X_1 \times X_2 \to Y_1 \times Y_2$. The wedge $Y_1 \vee Y_2$ depends on basepoints $\eta_i \in Y_i$ and can be defined as
$$Y_1 \vee Y_2 = (Y_1,\eta_1) \vee (Y_2,\eta_2) = \{(y_1,y_2) \in Y_1 \times Y_2 \mid y_1 = \eta_1 \text{ or } y_2 = \eta_2 \} .$$
By restricton of $p_1 \times p_2$ we then get a covering
$$\pi : (p_1 \times p_2)^{-1}(Y_1 \vee Y_2) \to Y_1 \vee Y_2$$
which has $n_1 \cdot n_2$ sheets if the $p_i$ have $n_i$ sheets. We have
$$(p_1 \times p_2)^{-1}(Y_1 \vee Y_2) = \{(x_1,x_2) \in X_1 \times X_2 \mid x_1 \in p_1^{-1}(\eta_1) \text{ or } x_2 \in p_2^{-1}(\eta_2) \}  \\ = X_1 \times p_2^{-1}(\eta_2) \cup p_1^{-1}(\eta_1) \times X_2 .$$
It is therefore obtained as a "multiple wedge" by taking the disjoint union $\bigsqcup _{x_2 \in p_2^{-1}(\eta_2)} X_1(x_2) \sqcup \bigsqcup _{x_1 \in p_1^{-1}(\eta_1)} X_2(x_1)$, where the  $X_1(x_2)$ and $X_2(x_1)$ are copies of $X_1$ and $X_2$, respectively, and identifying for all pairs $(x_1,x_2) \in p_1^{-1}(\eta_1) \times p_2^{-1}(\eta_2)$ the points $x_1 \in X_1(x_2)$ and $x_2 \in X_2(x_1)$.
Apply this gneral result to $p_1 = id : \Bbb RP^2 \to \Bbb RP^2$ and $p_2 : S^1 \to S^1, p_2(z) = z^3$. We get a covering with $3$ sheets and it is precisely the map you described.
