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The homotopy equivalence of suspension of a CW complex, Hatcher AT exercise $0.25$

1

$begingroup$

Exercise $0.25$ in Allen Hatcher's Algebraic Topology:

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, then it will come from definition.

general-topology cw-complexes

share|cite|improve this question

edited Mar 25 at 7:28

Andrews

asked Oct 30 '18 at 17:44

AndrewsAndrews

1,2962423

$endgroup$

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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
  
  

  
  
  
  


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  @JohnHughes Thanks for your time and help. I misunderstood the definition of $SX$.
  $endgroup$
  – Andrews
  Oct 31 '18 at 8:44
  

  
  
  
  


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  $begingroup$
  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
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
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  @ArnaudD. OK.Thanks.
  $endgroup$
  – Andrews
  Oct 31 '18 at 11:21
  

  
  
  
  


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  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
  

  
  
  
  

 | 
show 1 more comment

1

$begingroup$

Exercise $0.25$ in Allen Hatcher's Algebraic Topology:

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, then it will come from definition.

general-topology cw-complexes

share|cite|improve this question

edited Mar 25 at 7:28

Andrews

asked Oct 30 '18 at 17:44

AndrewsAndrews

1,2962423

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  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
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

Exercise $0.25$ in Allen Hatcher's Algebraic Topology:

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, then it will come from definition.

general-topology cw-complexes

share|cite|improve this question

edited Mar 25 at 7:28

Andrews

asked Oct 30 '18 at 17:44

AndrewsAndrews

1,2962423

$endgroup$

share|cite|improve this question

edited Mar 25 at 7:28

Andrews

asked Oct 30 '18 at 17:44

AndrewsAndrews

1,2962423

share|cite|improve this question

edited Mar 25 at 7:28

Andrews

asked Oct 30 '18 at 17:44

AndrewsAndrews

1,2962423

asked Oct 30 '18 at 17:44

AndrewsAndrews

1,2962423

asked Oct 30 '18 at 17:44

AndrewsAndrews

1,2962423