0
$\begingroup$

I have to answer the next question (for homework):

Find the generators for the fundamental group of the given spaces and write the curves mentioned as combination of the generators:

  1. The torus: C= the identity
  2. The cylinder: C= the top circumference
  3. The Möbius strip: C= the “boundary” of the strip.

Here’s what I understand. First, the fundamental grupo of the torus is isomorphic to $\mathbb{Z}\times \mathbb{Z}$. The generators for this group are (1,0) and (0,1). So the identity should be something like (1,0)*(0,1). I’m not sure what to do.

An example on what I’m asked to do (maybe with another surface or another curve) would be really helpful.

$\endgroup$
  • $\begingroup$ The identity on the torus isn't a curve. Maybe your professor means one of the inclusions $S^1\hookrightarrow S^1\times S^1$, either into one of the factors or diagonally? $\endgroup$ – Tyrone Oct 11 '17 at 18:57
  • $\begingroup$ The diagonal. The ones of the form (t,t) $\endgroup$ – user419934 Oct 11 '17 at 18:58
0
$\begingroup$

Note that any map into the product is uniquely determined by the composition with the projections $S^1\xrightarrow{f} S^1\times S^1\xrightarrow{pr_i}S^1$, writing $f=(pr_1\circ f,pr_2\circ f)$.This means for the diagonal we have $\Delta=(id_{S^1},id_{S^1})$. Now recall that we have the isomorphism $pr_{1*}\oplus pr_{2*}:\pi_1(S^1\times S^1)\cong\pi_1(S^1)\oplus\pi_1(S^1)$. In particular $pr_{1*}\oplus pr_{2*}([\Delta])=([id_{S^1}],[id_{S^1}])\in\pi_1(S^1)\oplus\pi_1(S^1)$. This means that in your notation $[\Delta]=(1,0)+(0,1)=(1,1)$.

For the cylinder, note that it deformation retracts onto the boundary circle. This gives you the generator and the curve. Similarly for the mobius strip, which deformation retracts onto the central circle (think what this implies the generator of $\pi_1$ is, and what the retraction does to the boundary in relation to this generator).

$\endgroup$
  • $\begingroup$ For the Möbius strip: the fundamental grpup is isomorphic to $\mathbb{Z}$. The boundary is a curve that surrounds the central circle twice. So C=2? $\endgroup$ – user419934 Oct 12 '17 at 0:02
  • $\begingroup$ That's the idea, but I think you might want to be a bit more explicit in terms of generators (which is what the question seemed to ask for) when writing it up. $\endgroup$ – Tyrone Oct 12 '17 at 4:07
  • $\begingroup$ I think I understand the instructions and with your example I'll complete the other two. Thank you very much. $\endgroup$ – user419934 Oct 12 '17 at 15:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy