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Different ways of identifying the boundaries of a disc with two holes
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I am reading Hatcher's Algebraic Topology and I have encountered a problem with exercise 1.2.13. The question goes as follows:
Let $Y$ be the space obtained from a disc with two holes by identifying each of the three boundary circles. There are only two essentially different ways of doing this each with a non-isomorphic fundamental group.
I understand that two different spaces come from whether you reverse the orientation of one of the identifications. However, my problem is, is there a nice way to realize these distinct spaces as CW complexes, preferably with a diagram?
Other questions regarding this exercise have been asked here and here.
general-topology algebraic-topology
asked Aug 1 at 19:41
Sam Hughes
1037
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I am reading Hatcher's Algebraic Topology and I have encountered a problem with exercise 1.2.13. The question goes as follows:
Let $Y$ be the space obtained from a disc with two holes by identifying each of the three boundary circles. There are only two essentially different ways of doing this each with a non-isomorphic fundamental group.
I understand that two different spaces come from whether you reverse the orientation of one of the identifications. However, my problem is, is there a nice way to realize these distinct spaces as CW complexes, preferably with a diagram?
Other questions regarding this exercise have been asked here and here.
general-topology algebraic-topology
asked Aug 1 at 19:41
Sam Hughes
1037
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am reading Hatcher's Algebraic Topology and I have encountered a problem with exercise 1.2.13. The question goes as follows:
Let $Y$ be the space obtained from a disc with two holes by identifying each of the three boundary circles. There are only two essentially different ways of doing this each with a non-isomorphic fundamental group.
I understand that two different spaces come from whether you reverse the orientation of one of the identifications. However, my problem is, is there a nice way to realize these distinct spaces as CW complexes, preferably with a diagram?
Other questions regarding this exercise have been asked here and here.
general-topology algebraic-topology
asked Aug 1 at 19:41
Sam Hughes
1037
I am reading Hatcher's Algebraic Topology and I have encountered a problem with exercise 1.2.13. The question goes as follows:
Let $Y$ be the space obtained from a disc with two holes by identifying each of the three boundary circles. There are only two essentially different ways of doing this each with a non-isomorphic fundamental group.
I understand that two different spaces come from whether you reverse the orientation of one of the identifications. However, my problem is, is there a nice way to realize these distinct spaces as CW complexes, preferably with a diagram?
Other questions regarding this exercise have been asked here and here.
general-topology algebraic-topology
asked Aug 1 at 19:41
Sam Hughes
1037
asked Aug 1 at 19:41
Sam Hughes
1037
asked Aug 1 at 19:41
Sam Hughes
1037
asked Aug 1 at 19:41
Sam Hughes
1037
1037
add a comment |Â
add a comment |Â
1 Answer
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Begin by drawing the disk with two holes cut out. Give this a cell complex structure with three 0-cells, five 1-cells, and one 2-cell. Going from this space to the space $Y,$ you must identify the three 0-cells and two of the five 1-cells. Let $a$ be the 0-cell of $Y$; let $c,$ $lambda,$ and $ell$ be the 1-cells of $Y$; and let $P$ be the 2-cell of $Y.$ Our diagram should look similar to the following.
Computing the cellular homology of $Y$ now amounts to simply writing down the matrices of the cellular boundary maps with 0s, 1s, and -1s, and calculating kernels and cokernels.
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Begin by drawing the disk with two holes cut out. Give this a cell complex structure with three 0-cells, five 1-cells, and one 2-cell. Going from this space to the space $Y,$ you must identify the three 0-cells and two of the five 1-cells. Let $a$ be the 0-cell of $Y$; let $c,$ $lambda,$ and $ell$ be the 1-cells of $Y$; and let $P$ be the 2-cell of $Y.$ Our diagram should look similar to the following.
Computing the cellular homology of $Y$ now amounts to simply writing down the matrices of the cellular boundary maps with 0s, 1s, and -1s, and calculating kernels and cokernels.
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
add a comment |Â
up vote
2
down vote
accepted
Begin by drawing the disk with two holes cut out. Give this a cell complex structure with three 0-cells, five 1-cells, and one 2-cell. Going from this space to the space $Y,$ you must identify the three 0-cells and two of the five 1-cells. Let $a$ be the 0-cell of $Y$; let $c,$ $lambda,$ and $ell$ be the 1-cells of $Y$; and let $P$ be the 2-cell of $Y.$ Our diagram should look similar to the following.
Computing the cellular homology of $Y$ now amounts to simply writing down the matrices of the cellular boundary maps with 0s, 1s, and -1s, and calculating kernels and cokernels.
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Begin by drawing the disk with two holes cut out. Give this a cell complex structure with three 0-cells, five 1-cells, and one 2-cell. Going from this space to the space $Y,$ you must identify the three 0-cells and two of the five 1-cells. Let $a$ be the 0-cell of $Y$; let $c,$ $lambda,$ and $ell$ be the 1-cells of $Y$; and let $P$ be the 2-cell of $Y.$ Our diagram should look similar to the following.
Computing the cellular homology of $Y$ now amounts to simply writing down the matrices of the cellular boundary maps with 0s, 1s, and -1s, and calculating kernels and cokernels.
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
Begin by drawing the disk with two holes cut out. Give this a cell complex structure with three 0-cells, five 1-cells, and one 2-cell. Going from this space to the space $Y,$ you must identify the three 0-cells and two of the five 1-cells. Let $a$ be the 0-cell of $Y$; let $c,$ $lambda,$ and $ell$ be the 1-cells of $Y$; and let $P$ be the 2-cell of $Y.$ Our diagram should look similar to the following.
Computing the cellular homology of $Y$ now amounts to simply writing down the matrices of the cellular boundary maps with 0s, 1s, and -1s, and calculating kernels and cokernels.
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
answered Aug 2 at 16:21
Dylan_Carlo_Beck
342210
342210
add a comment |Â
add a comment |Â
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