Find the fundamental group of the following spaces
$begingroup$
- Find the fundamental groups of the following spaces. In each case they
can be built up from cyclic groups by free products and direct products.
(a) The space obtained from two copies of the torus $S^1times S^1$ by identifying the simple closed curve $S^1times{1}$ on the first copy of the torus with the simple closed curve $S^1 times {1}$ on the second copy of the
torus.
(b) The space obtained from a torus $S^1 times S^1$ by joining two distinct
points $a$ and $b$ on it by an arc which meets the torus only in its two endpoints $a$ and $b$
My trouble is with the second one, I don't know how to apply Van Kampen's theorem to this space.
algebraic-topology
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
$endgroup$
add a comment |
$begingroup$
- Find the fundamental groups of the following spaces. In each case they
can be built up from cyclic groups by free products and direct products.
(a) The space obtained from two copies of the torus $S^1times S^1$ by identifying the simple closed curve $S^1times{1}$ on the first copy of the torus with the simple closed curve $S^1 times {1}$ on the second copy of the
torus.
(b) The space obtained from a torus $S^1 times S^1$ by joining two distinct
points $a$ and $b$ on it by an arc which meets the torus only in its two endpoints $a$ and $b$
My trouble is with the second one, I don't know how to apply Van Kampen's theorem to this space.
algebraic-topology
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
$endgroup$
1
$begingroup$
Well you know the fundamental group for the torus if you did part (a), so find two open sets in the space described in part (b) and apply the theorem again.
$endgroup$
– DKS
Dec 10 '18 at 4:01
$begingroup$
Interesting. I know the fundamental group of the torus is $mathbb{Z}timesmathbb{Z}$ from outside the problem, but I don't even recognize how part (a) relates to part (b).
$endgroup$
– Adam Cartisano
Dec 10 '18 at 14:23
$begingroup$
For (b) you can take a nbhd of the circle containing $a$ and $b$, and then a nbhd of the torus. You can also homotope the space to a wedge of the torus and circle.
$endgroup$
– Hempelicious
Dec 10 '18 at 19:56
$begingroup$
I didn't know if I was allowed to consider the two points of intersection as the same point via a homotopy. In that case, the solution would be $mathbb{Z}timesmathbb{Z}$ free product with $mathbb{Z}$?
$endgroup$
– Adam Cartisano
Dec 10 '18 at 20:24
$begingroup$
Yes. Now try to work it out with the van Kampen theorem!
$endgroup$
– Hempelicious
Dec 11 '18 at 7:34
add a comment |
$begingroup$
- Find the fundamental groups of the following spaces. In each case they
can be built up from cyclic groups by free products and direct products.
(a) The space obtained from two copies of the torus $S^1times S^1$ by identifying the simple closed curve $S^1times{1}$ on the first copy of the torus with the simple closed curve $S^1 times {1}$ on the second copy of the
torus.
(b) The space obtained from a torus $S^1 times S^1$ by joining two distinct
points $a$ and $b$ on it by an arc which meets the torus only in its two endpoints $a$ and $b$
My trouble is with the second one, I don't know how to apply Van Kampen's theorem to this space.
algebraic-topology
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
$endgroup$
- Find the fundamental groups of the following spaces. In each case they
can be built up from cyclic groups by free products and direct products.
(a) The space obtained from two copies of the torus $S^1times S^1$ by identifying the simple closed curve $S^1times{1}$ on the first copy of the torus with the simple closed curve $S^1 times {1}$ on the second copy of the
torus.
(b) The space obtained from a torus $S^1 times S^1$ by joining two distinct
points $a$ and $b$ on it by an arc which meets the torus only in its two endpoints $a$ and $b$
My trouble is with the second one, I don't know how to apply Van Kampen's theorem to this space.
algebraic-topology
algebraic-topology
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
asked Dec 10 '18 at 3:57
Adam CartisanoAdam Cartisano
1764
1764
1
$begingroup$
Well you know the fundamental group for the torus if you did part (a), so find two open sets in the space described in part (b) and apply the theorem again.
$endgroup$
– DKS
Dec 10 '18 at 4:01
$begingroup$
Interesting. I know the fundamental group of the torus is $mathbb{Z}timesmathbb{Z}$ from outside the problem, but I don't even recognize how part (a) relates to part (b).
$endgroup$
– Adam Cartisano
Dec 10 '18 at 14:23
$begingroup$
For (b) you can take a nbhd of the circle containing $a$ and $b$, and then a nbhd of the torus. You can also homotope the space to a wedge of the torus and circle.
$endgroup$
– Hempelicious
Dec 10 '18 at 19:56
$begingroup$
I didn't know if I was allowed to consider the two points of intersection as the same point via a homotopy. In that case, the solution would be $mathbb{Z}timesmathbb{Z}$ free product with $mathbb{Z}$?
$endgroup$
– Adam Cartisano
Dec 10 '18 at 20:24
$begingroup$
Yes. Now try to work it out with the van Kampen theorem!
$endgroup$
– Hempelicious
Dec 11 '18 at 7:34
add a comment |
1
$begingroup$
Well you know the fundamental group for the torus if you did part (a), so find two open sets in the space described in part (b) and apply the theorem again.
$endgroup$
– DKS
Dec 10 '18 at 4:01
$begingroup$
Interesting. I know the fundamental group of the torus is $mathbb{Z}timesmathbb{Z}$ from outside the problem, but I don't even recognize how part (a) relates to part (b).
$endgroup$
– Adam Cartisano
Dec 10 '18 at 14:23
$begingroup$
For (b) you can take a nbhd of the circle containing $a$ and $b$, and then a nbhd of the torus. You can also homotope the space to a wedge of the torus and circle.
$endgroup$
– Hempelicious
Dec 10 '18 at 19:56
$begingroup$
I didn't know if I was allowed to consider the two points of intersection as the same point via a homotopy. In that case, the solution would be $mathbb{Z}timesmathbb{Z}$ free product with $mathbb{Z}$?
$endgroup$
– Adam Cartisano
Dec 10 '18 at 20:24
$begingroup$
Yes. Now try to work it out with the van Kampen theorem!
$endgroup$
– Hempelicious
Dec 11 '18 at 7:34
1
1
$begingroup$
Well you know the fundamental group for the torus if you did part (a), so find two open sets in the space described in part (b) and apply the theorem again.
$endgroup$
– DKS
Dec 10 '18 at 4:01
$begingroup$
Well you know the fundamental group for the torus if you did part (a), so find two open sets in the space described in part (b) and apply the theorem again.
$endgroup$
– DKS
Dec 10 '18 at 4:01
$begingroup$
Interesting. I know the fundamental group of the torus is $mathbb{Z}timesmathbb{Z}$ from outside the problem, but I don't even recognize how part (a) relates to part (b).
$endgroup$
– Adam Cartisano
Dec 10 '18 at 14:23
$begingroup$
Interesting. I know the fundamental group of the torus is $mathbb{Z}timesmathbb{Z}$ from outside the problem, but I don't even recognize how part (a) relates to part (b).
$endgroup$
– Adam Cartisano
Dec 10 '18 at 14:23
$begingroup$
For (b) you can take a nbhd of the circle containing $a$ and $b$, and then a nbhd of the torus. You can also homotope the space to a wedge of the torus and circle.
$endgroup$
– Hempelicious
Dec 10 '18 at 19:56
$begingroup$
For (b) you can take a nbhd of the circle containing $a$ and $b$, and then a nbhd of the torus. You can also homotope the space to a wedge of the torus and circle.
$endgroup$
– Hempelicious
Dec 10 '18 at 19:56
$begingroup$
I didn't know if I was allowed to consider the two points of intersection as the same point via a homotopy. In that case, the solution would be $mathbb{Z}timesmathbb{Z}$ free product with $mathbb{Z}$?
$endgroup$
– Adam Cartisano
Dec 10 '18 at 20:24
$begingroup$
I didn't know if I was allowed to consider the two points of intersection as the same point via a homotopy. In that case, the solution would be $mathbb{Z}timesmathbb{Z}$ free product with $mathbb{Z}$?
$endgroup$
– Adam Cartisano
Dec 10 '18 at 20:24
$begingroup$
Yes. Now try to work it out with the van Kampen theorem!
$endgroup$
– Hempelicious
Dec 11 '18 at 7:34
$begingroup$
Yes. Now try to work it out with the van Kampen theorem!
$endgroup$
– Hempelicious
Dec 11 '18 at 7:34
add a comment |
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$begingroup$
Well you know the fundamental group for the torus if you did part (a), so find two open sets in the space described in part (b) and apply the theorem again.
$endgroup$
– DKS
Dec 10 '18 at 4:01
$begingroup$
Interesting. I know the fundamental group of the torus is $mathbb{Z}timesmathbb{Z}$ from outside the problem, but I don't even recognize how part (a) relates to part (b).
$endgroup$
– Adam Cartisano
Dec 10 '18 at 14:23
$begingroup$
For (b) you can take a nbhd of the circle containing $a$ and $b$, and then a nbhd of the torus. You can also homotope the space to a wedge of the torus and circle.
$endgroup$
– Hempelicious
Dec 10 '18 at 19:56
$begingroup$
I didn't know if I was allowed to consider the two points of intersection as the same point via a homotopy. In that case, the solution would be $mathbb{Z}timesmathbb{Z}$ free product with $mathbb{Z}$?
$endgroup$
– Adam Cartisano
Dec 10 '18 at 20:24
$begingroup$
Yes. Now try to work it out with the van Kampen theorem!
$endgroup$
– Hempelicious
Dec 11 '18 at 7:34