Hatcher Ex $0.25$, the homotopy equivalence of suspension of a CW complex
1
$begingroup$
Exercise $0.25$:
If $X$ is a CW complex with components $X_alpha$, show that the suspension $SX$ is homotopy equivalent to $Ylor_alpha SX_alpha$ for some graph $Y$.
In the case that $X$ is a finite graph, show that $SX$ is homotopy equivalent to a wedge sum of circles and $2$-spheres.
Edit: It's not hard actually, from the basic examples to find what graph $Y$ is.
general-topology algebraic-topology cw-complexes
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
$endgroup$
add a comment |
1
$begingroup$
Exercise $0.25$:
If $X$ is a CW complex with components $X_alpha$, show that the suspension $SX$ is homotopy equivalent to $Ylor_alpha SX_alpha$ for some graph $Y$.
In the case that $X$ is a finite graph, show that $SX$ is homotopy equivalent to a wedge sum of circles and $2$-spheres.
Edit: It's not hard actually, from the basic examples to find what graph $Y$ is.
general-topology algebraic-topology cw-complexes
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
$endgroup$
$begingroup$
Consider the case where $X$ is two points -- say $+1$ and $-1$ on the number line. The suspension adds two more points, one above the line and one below, and connects each to the points of $X$, yielding a diamond, which is topologically a circle. So it's certainly possible. You might want to think through the case where $X$ consists of just 3 points on the number line. Unrelated: What does that subscript "alpha" on the join $U vee_alpha SX_alpha$ mean?
$endgroup$
– John Hughes
Oct 30 '18 at 17:57
$begingroup$
@JohnHughes Thanks for your time and help. I misunderstood the definition of $SX$.
$endgroup$
– Andrews
Oct 31 '18 at 8:44
2
$begingroup$
If you have found your answer, you can write an answer to your own question (and even accept it after a delay). It could be useful for others. In any case, avoid modifying the title like you did.
$endgroup$
– Arnaud D.
Oct 31 '18 at 11:15
$begingroup$
@ArnaudD. OK.Thanks.
$endgroup$
– Andrews
Oct 31 '18 at 11:21
1
$begingroup$
I agree,,,writing up your answer (without completely solving the problem, which would lessen the value of Hatcher's book to future students) would be a great way to give back to MSE (and to solidify your own understanding of the point on which you were confused).
$endgroup$
– John Hughes
Oct 31 '18 at 15:40
add a comment |
1
1
1
$begingroup$
Exercise $0.25$:
If $X$ is a CW complex with components $X_alpha$, show that the suspension $SX$ is homotopy equivalent to $Ylor_alpha SX_alpha$ for some graph $Y$.
In the case that $X$ is a finite graph, show that $SX$ is homotopy equivalent to a wedge sum of circles and $2$-spheres.
Edit: It's not hard actually, from the basic examples to find what graph $Y$ is.
general-topology algebraic-topology cw-complexes
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
$endgroup$
Exercise $0.25$:
If $X$ is a CW complex with components $X_alpha$, show that the suspension $SX$ is homotopy equivalent to $Ylor_alpha SX_alpha$ for some graph $Y$.
In the case that $X$ is a finite graph, show that $SX$ is homotopy equivalent to a wedge sum of circles and $2$-spheres.
Edit: It's not hard actually, from the basic examples to find what graph $Y$ is.
general-topology algebraic-topology cw-complexes
general-topology algebraic-topology cw-complexes
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
edited Dec 5 '18 at 22:18
Andrews
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
asked Oct 30 '18 at 17:44
AndrewsAndrews
3791317
3791317
$begingroup$
Consider the case where $X$ is two points -- say $+1$ and $-1$ on the number line. The suspension adds two more points, one above the line and one below, and connects each to the points of $X$, yielding a diamond, which is topologically a circle. So it's certainly possible. You might want to think through the case where $X$ consists of just 3 points on the number line. Unrelated: What does that subscript "alpha" on the join $U vee_alpha SX_alpha$ mean?
$endgroup$
– John Hughes
Oct 30 '18 at 17:57
$begingroup$
@JohnHughes Thanks for your time and help. I misunderstood the definition of $SX$.
$endgroup$
– Andrews
Oct 31 '18 at 8:44
2
$begingroup$
If you have found your answer, you can write an answer to your own question (and even accept it after a delay). It could be useful for others. In any case, avoid modifying the title like you did.
$endgroup$
– Arnaud D.
Oct 31 '18 at 11:15
$begingroup$
@ArnaudD. OK.Thanks.
$endgroup$
– Andrews
Oct 31 '18 at 11:21
1
$begingroup$
I agree,,,writing up your answer (without completely solving the problem, which would lessen the value of Hatcher's book to future students) would be a great way to give back to MSE (and to solidify your own understanding of the point on which you were confused).
$endgroup$
– John Hughes
Oct 31 '18 at 15:40
add a comment |
$begingroup$
Consider the case where $X$ is two points -- say $+1$ and $-1$ on the number line. The suspension adds two more points, one above the line and one below, and connects each to the points of $X$, yielding a diamond, which is topologically a circle. So it's certainly possible. You might want to think through the case where $X$ consists of just 3 points on the number line. Unrelated: What does that subscript "alpha" on the join $U vee_alpha SX_alpha$ mean?
$endgroup$
– John Hughes
Oct 30 '18 at 17:57
$begingroup$
@JohnHughes Thanks for your time and help. I misunderstood the definition of $SX$.
$endgroup$
– Andrews
Oct 31 '18 at 8:44
2
$begingroup$
If you have found your answer, you can write an answer to your own question (and even accept it after a delay). It could be useful for others. In any case, avoid modifying the title like you did.
$endgroup$
– Arnaud D.
Oct 31 '18 at 11:15
$begingroup$
@ArnaudD. OK.Thanks.
$endgroup$
– Andrews
Oct 31 '18 at 11:21
1
$begingroup$
I agree,,,writing up your answer (without completely solving the problem, which would lessen the value of Hatcher's book to future students) would be a great way to give back to MSE (and to solidify your own understanding of the point on which you were confused).
$endgroup$
– John Hughes
Oct 31 '18 at 15:40
$begingroup$
Consider the case where $X$ is two points -- say $+1$ and $-1$ on the number line. The suspension adds two more points, one above the line and one below, and connects each to the points of $X$, yielding a diamond, which is topologically a circle. So it's certainly possible. You might want to think through the case where $X$ consists of just 3 points on the number line. Unrelated: What does that subscript "alpha" on the join $U vee_alpha SX_alpha$ mean?
$endgroup$
– John Hughes
Oct 30 '18 at 17:57
$begingroup$
Consider the case where $X$ is two points -- say $+1$ and $-1$ on the number line. The suspension adds two more points, one above the line and one below, and connects each to the points of $X$, yielding a diamond, which is topologically a circle. So it's certainly possible. You might want to think through the case where $X$ consists of just 3 points on the number line. Unrelated: What does that subscript "alpha" on the join $U vee_alpha SX_alpha$ mean?
$endgroup$
– John Hughes
Oct 30 '18 at 17:57
$begingroup$
@JohnHughes Thanks for your time and help. I misunderstood the definition of $SX$.
$endgroup$
– Andrews
Oct 31 '18 at 8:44
$begingroup$
@JohnHughes Thanks for your time and help. I misunderstood the definition of $SX$.
$endgroup$
– Andrews
Oct 31 '18 at 8:44
2
2
$begingroup$
If you have found your answer, you can write an answer to your own question (and even accept it after a delay). It could be useful for others. In any case, avoid modifying the title like you did.
$endgroup$
– Arnaud D.
Oct 31 '18 at 11:15
$begingroup$
If you have found your answer, you can write an answer to your own question (and even accept it after a delay). It could be useful for others. In any case, avoid modifying the title like you did.
$endgroup$
– Arnaud D.
Oct 31 '18 at 11:15
$begingroup$
@ArnaudD. OK.Thanks.
$endgroup$
– Andrews
Oct 31 '18 at 11:21
$begingroup$
@ArnaudD. OK.Thanks.
$endgroup$
– Andrews
Oct 31 '18 at 11:21
1
1
$begingroup$
I agree,,,writing up your answer (without completely solving the problem, which would lessen the value of Hatcher's book to future students) would be a great way to give back to MSE (and to solidify your own understanding of the point on which you were confused).
$endgroup$
– John Hughes
Oct 31 '18 at 15:40
$begingroup$
I agree,,,writing up your answer (without completely solving the problem, which would lessen the value of Hatcher's book to future students) would be a great way to give back to MSE (and to solidify your own understanding of the point on which you were confused).
$endgroup$
– John Hughes
Oct 31 '18 at 15:40
add a comment |
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$begingroup$
Consider the case where $X$ is two points -- say $+1$ and $-1$ on the number line. The suspension adds two more points, one above the line and one below, and connects each to the points of $X$, yielding a diamond, which is topologically a circle. So it's certainly possible. You might want to think through the case where $X$ consists of just 3 points on the number line. Unrelated: What does that subscript "alpha" on the join $U vee_alpha SX_alpha$ mean?
$endgroup$
– John Hughes
Oct 30 '18 at 17:57
$begingroup$
@JohnHughes Thanks for your time and help. I misunderstood the definition of $SX$.
$endgroup$
– Andrews
Oct 31 '18 at 8:44
2
$begingroup$
If you have found your answer, you can write an answer to your own question (and even accept it after a delay). It could be useful for others. In any case, avoid modifying the title like you did.
$endgroup$
– Arnaud D.
Oct 31 '18 at 11:15
$begingroup$
@ArnaudD. OK.Thanks.
$endgroup$
– Andrews
Oct 31 '18 at 11:21
1
$begingroup$
I agree,,,writing up your answer (without completely solving the problem, which would lessen the value of Hatcher's book to future students) would be a great way to give back to MSE (and to solidify your own understanding of the point on which you were confused).
$endgroup$
– John Hughes
Oct 31 '18 at 15:40