Looking for instructive examples on the difference between homology and homotopy, I found here the following example:

Example: Consider an oriented loop separating a genus $2$ surface into two genus $1$ punctured surfaces. This loop is nontrivial in the fundamental group, but is trivial in homology, i.e. it is homologous to zero.

I don't understand this. I'm imagining a double torus with a rubber band around the "connection of the two tori", but this curve is contractible since I can just pull the rubber band off around either hole. What is meant in the example?

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    $\begingroup$ No, you can't pull the rubber band off around either hole. It would leave the surface and span across the hole in the process. $\endgroup$ – Daniel Fischer Nov 20 '14 at 16:54
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    $\begingroup$ You cannot contract it over the hole. Remember, the rubber band needs to be in contact with the torus all along. It cannot at any point be suspended over the hole. Algebraically, that curve is a commutator in the fundamental group, and since the homology group of dimension one is the abelianisation of the fundamental group, it becomes the identity element. $\endgroup$ – Arthur Nov 20 '14 at 16:54
  • $\begingroup$ Ah! I completely overlooked the band leaving the surface! Good thing I made this mistake relatively early. Thanks. Perhaps one of you should post an answer so that I have something to accept. $\endgroup$ – user153312 Nov 20 '14 at 17:07
  • $\begingroup$ @Daniel, Arthur, what do the homology classes of the double torus look like? Are they generated by this loop along with the two "usual" ones on the torus? Why is this loop homologous to zero? $\endgroup$ – user153312 Nov 20 '14 at 20:46

This loop bounds a subsurface (torus with one hole) of genus 2 surface. We know that non-trivial elements of first homology group is 'cycles that are not boundary'. On the other hand since this loop does not bound a disc it is non-trivial on fundemental group.It can be also checked algebraically by drawing planar diagram of genus two surface and see that which element the loop correspond to in fundemental group.

  • $\begingroup$ How can I see this loop bounds one of the tori? $\endgroup$ – user153312 Nov 20 '14 at 18:15
  • $\begingroup$ Actually torus is a closed manifold therefore it has no boundary. If you cut the genus two surface from the loop you have two torus with one hole and the loop bounds these holes. $\endgroup$ – camsilbira Nov 20 '14 at 18:20

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