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The problem is in the picture.enter image description here

My question is How could $M_g$ retract onto $C'$?That seems impossible.Thank you.

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Recall that a retract is not a deformation retraction. So let's visualize this on a torus, and you'll see how this works.

View the torus as a square with the normal identifications, and suppose that our C' are the vertical edges (which are identified). Then simply split the square down the middle, and push the left half to the left edge and the right half to the right edge. This corresponds to cutting our torus into a cylinder, and then squishing it into the circle.

This might sound very poor, but not that the squishing reunites the parts that were split. I suspect if you draw a few neighborhoods and look how they do, in fact, stay close, you'll become convinced that this is a retract.

Can you generalize this to higher genus?

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    You should follow the hint. Abelianize the fundamental group.2012-04-12