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Hello there i am having trouble to determine isomorphisms of the following fundamental groups:

1) the torus $T$ with a removed point.

2) $\mathbb{R}^3$ with nonnegative axes

3) $S^1 \cup (\mathbb{R} \times 0 )$

i am using deformation retractions most of the time to find a solution for such problems. But for these 3 particular problems i can't seem to figure out the required retraction :(.

Are there any other theorems/definitions which can also determine isomorphisms of particular fundamental groups?

Kees Til

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    $\begingroup$ What do you mean "with nonnegative axes"? $\endgroup$ Commented May 10, 2014 at 17:20
  • $\begingroup$ @ThomasAndrews he means remove the sets $\{(x,0,0)|x\geq 0\}$, $\{(0,y,0)|y\geq 0\}$ and $\{(0,0,z)|z\geq 0\}$ $\endgroup$
    – Seth
    Commented May 10, 2014 at 17:46

1 Answer 1

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1) the Torus with a point removed deformation retracts onto the figure 8. One way to see this is to use the square with edges identified. Remove a point from the middle and then imagine stretched the point out to the edges. The edges are a wedge of two circles.

2) $\mathbb{R}^3$ with the nonnegative axis removed deformation retracts onto a "3-pipe space" enter image description here

This space deformation retracts to a figure 8 by pushing the pipes in to the center and kind of twisting the top pipe onto the others.

3) $S^1 \cup (\mathbb{R}\times 0)$ deformation retracts to a circle with a line segment connecting two antipodal points. This is homotopic to the figure 8 by contracting the line segment to a point (or you can use van kampens to compute the fundamental group).

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  • $\begingroup$ i don't fullt understand 1, what do you do after streching the hole to the edges? And how is it possible to deform the top of the pipes in a continuous way? $\endgroup$
    – Kees Til
    Commented May 10, 2014 at 18:00
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    $\begingroup$ @KeesTil For 1 once you deformation retract onto the edges, notice that the 4 corners are identified. So the space is just a point with some loops attached. Specifically 2 loops since the 4 edges are identified in pairs. For 2 it is kind of hard to explain - sorry if this part isn't satisfactory. Try to imagine pushing the top pipe down and then rotating it 90 degrees downwards (leaving the front point fixed, pushing the back part down). You get 2 pipes that share a hole and are attached by a semi circle. Then push each pipe into a circle. You get the figure 8 space. $\endgroup$
    – Seth
    Commented May 10, 2014 at 18:03

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