fundamental groups Watch

jdavi
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Please can someone explain to me why a torus minus a point is homotopy equivalent to the figure 8 curve, i.e. the wedge sum of two circles.

More generally can someone explain why a torus minus n points is homotopy equivalent to the wedge sum of n+1 circles.

Any help would be much appreciated!
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nuodai
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For the case with one point removed, consider stretching the hole so that it goes around a circular cross-section of the torus except at a point. That is, you have a doughnut with one slice in it, except that it's still hanging on at a point. Now stretch the resulting slit until it goes around the rest of the torus. What you're left with is a "horizontal" circle and a "vertical" circle which are attached at the point, i.e. the wedge sum of two circles.

Think about how you can generalise this to n points removed.
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SsEe
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(Original post by jdavi)
Please can someone explain to me why a torus minus a point is homotopy equivalent to the figure 8 curve, i.e. the wedge sum of two circles.

More generally can someone explain why a torus minus n points is homotopy equivalent to the wedge sum of n+1 circles.

Any help would be much appreciated!
In words? Pff. Um... Think about a square with the edges identified. Expand the hole so all you're left with is the circumference of the square (with identification). Then do the identification and see what you get.
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