Let $X_2$ and $X_3$ denote closed, oriented surfaces of genus $2$ and $3$ respectively.

  1. What is a specific degree $2$ covering map $g$ from $X_3$ to $X_2$?

Let $G: X_3 \to X_3 \times X_2$ be the graph of that specific $g$, i.e. the map taking each point $x$ to $(x, g(x))$.

  1. What maps does $G$ induce on homology?
  2. What maps does $G$ induce on $\pi_1$?
  • 1
    $\begingroup$ You should include your thoughts about the problem. Also, I am not sure what "specific" refers to. For instance, do you have hard time constructing explicit epimorphisms from $\pi_1(X_2)$ to ${\mathbb Z}/2$? Are you aware of the Kunneth formula? $\endgroup$ – Moishe Kohan Oct 24 '16 at 21:05

To answer the first question, visualize $X_3$ as being three connected tori laying in a line (as apposed to connected like a flower). Look at a generating loop $\hat B_2$ of the middle torus, and send that loop $\hat B_2$ to the loop $2\hat B_2$. By doing this you are wrapping the middle torus over itself two-fold, and you are associated the two outer tori with each other.

Image of X_3 to X_2

Image found at inspirehep.net/record/733242/files/web2.png.

In this picture, you are coilling that middle loop $B_2$ around itself twice, which will send $\hat{A}_2$ onto $i\hat{A}_2$ and will send each of $\hat{A}_1$ and $\hat{B}_1$ in the left-most torus onto $i\hat{A}_1$ and $i\hat{B}_1$ in the right-most torus. Maybe think of it like closing a pocket-knife where the middle torus is the hinge.

Since the loops of the tori are really indistinguishable, you could also have folded the loop $\hat A_2$ over itself twice instead. Then you could visualize the map as like closing a thigh-master.


Image found at site.unbeatablesale.com/img097/rf026b.gif


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