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A problem in Wedge product of topological spaces

Why do we use the smash product in the category of based topological spaces?Hatcher Algebraic Topology 0.24Wedge Sum Embedding with InclusionsUnderstanding smash product of discrete space.Smash product of two pointed spacesInjectivity of $[(S^n, ;x_o), ;(S^n, ;x_o)]^circ longrightarrow [S^n, ;S^n]$Wedge Sum ExampleFixed point of a map of wedge productDefining the Smash product$X/Awedge Y/B= (Xtimes Y)/(Xtimes Bcup Atimes Y)$

4

$begingroup$

I am studying algebraic topology and I am a beginner in it. I want to prove that if $ (X, x_0) sim
(Y, y_0) $
and
$(Z, z_0) sim (W , w_0) $ then $ Xvee Z sim Yvee W .$ X,Y, Z, W are top spaces

I have just an intuition that wedge product seems something like glueing two spaces together in a point. I don't know how can I start to think about it and solve it. Any help would be great thanks.

algebraic-topology homotopy-theory

share|cite|improve this question

edited Jun 21 at 15:20

Eric Wofsey

201k1414 gold badges234234 silver badges365365 bronze badges

asked Jun 21 at 11:22

Fatemeh SafinejadFatemeh Safinejad

9666 bronze badges

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is most easily done by mapping properties of wedges and quotients.
  $endgroup$
  – Randall
  Jun 21 at 11:40
  

  
  
  
  

add a comment | 

4

$begingroup$

I am studying algebraic topology and I am a beginner in it. I want to prove that if $ (X, x_0) sim
(Y, y_0) $
and
$(Z, z_0) sim (W , w_0) $ then $ Xvee Z sim Yvee W .$ X,Y, Z, W are top spaces

I have just an intuition that wedge product seems something like glueing two spaces together in a point. I don't know how can I start to think about it and solve it. Any help would be great thanks.

algebraic-topology homotopy-theory

share|cite|improve this question

edited Jun 21 at 15:20

Eric Wofsey

201k1414 gold badges234234 silver badges365365 bronze badges

asked Jun 21 at 11:22

Fatemeh SafinejadFatemeh Safinejad

9666 bronze badges

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is most easily done by mapping properties of wedges and quotients.
  $endgroup$
  – Randall
  Jun 21 at 11:40
  

  
  
  
  

add a comment | 

$begingroup$

I am studying algebraic topology and I am a beginner in it. I want to prove that if $ (X, x_0) sim
(Y, y_0) $
and
$(Z, z_0) sim (W , w_0) $ then $ Xvee Z sim Yvee W .$ X,Y, Z, W are top spaces

I have just an intuition that wedge product seems something like glueing two spaces together in a point. I don't know how can I start to think about it and solve it. Any help would be great thanks.

algebraic-topology homotopy-theory

share|cite|improve this question

edited Jun 21 at 15:20

Eric Wofsey

201k1414 gold badges234234 silver badges365365 bronze badges

asked Jun 21 at 11:22

Fatemeh SafinejadFatemeh Safinejad

9666 bronze badges

$endgroup$

share|cite|improve this question

edited Jun 21 at 15:20

Eric Wofsey

201k1414 gold badges234234 silver badges365365 bronze badges

asked Jun 21 at 11:22

Fatemeh SafinejadFatemeh Safinejad

9666 bronze badges

share|cite|improve this question

edited Jun 21 at 15:20

Eric Wofsey

201k1414 gold badges234234 silver badges365365 bronze badges

asked Jun 21 at 11:22

Fatemeh SafinejadFatemeh Safinejad

9666 bronze badges

edited Jun 21 at 15:20

Eric Wofsey

201k1414 gold badges234234 silver badges365365 bronze badges

edited Jun 21 at 15:20

Eric Wofsey

201k1414 gold badges234234 silver badges365365 bronze badges

asked Jun 21 at 11:22

Fatemeh SafinejadFatemeh Safinejad

9666 bronze badges

asked Jun 21 at 11:22

Fatemeh SafinejadFatemeh Safinejad

9666 bronze badges

1 Answer
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oldest

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7

$begingroup$

Generally speaking, (particularly if you're new to algebraic topology), a good approach can be we actually write down the maps which give you a homotopy equivalence of spaces. I will give you a few hints to get you going.

HINTS:

• You start by assuming that $(X, x_0) sim (Y, y_0)$ and $(Z, z_0)
  sim (W , w_0)$. This is extremely important. That tells you that you have some maps back and
  forth which satisfy a certain property to do with their composition
  being homotopy equivalent to the identity. Keep these maps, and the homotopies which give you your homotopy equivalences, in mind.




• You can use these maps back and forth to build yourself some maps back and forth on the wedges of spaces $Xvee Z leftrightarrow Yvee W$.




• You then want to show that the composition of these maps is homotopic to the identity on both spaces - the definition of the spaces being homotopy equivalent. To do that, you're going to want to use the homotopies from earlier, which give you that your spaces are homotopy equivalent.

The above is the bare bones of what I imagine is the argument you'll want to run. Give it a shot, and then if you want more details please comment and I can try to say a little more.

share|cite|improve this answer

answered Jun 21 at 12:01

MattMatt

2,77888 silver badges2121 bronze badges

$endgroup$

add a comment | 

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7

$begingroup$

Generally speaking, (particularly if you're new to algebraic topology), a good approach can be we actually write down the maps which give you a homotopy equivalence of spaces. I will give you a few hints to get you going.

HINTS:

• You start by assuming that $(X, x_0) sim (Y, y_0)$ and $(Z, z_0)
  sim (W , w_0)$. This is extremely important. That tells you that you have some maps back and
  forth which satisfy a certain property to do with their composition
  being homotopy equivalent to the identity. Keep these maps, and the homotopies which give you your homotopy equivalences, in mind.




• You can use these maps back and forth to build yourself some maps back and forth on the wedges of spaces $Xvee Z leftrightarrow Yvee W$.




• You then want to show that the composition of these maps is homotopic to the identity on both spaces - the definition of the spaces being homotopy equivalent. To do that, you're going to want to use the homotopies from earlier, which give you that your spaces are homotopy equivalent.

The above is the bare bones of what I imagine is the argument you'll want to run. Give it a shot, and then if you want more details please comment and I can try to say a little more.

share|cite|improve this answer

answered Jun 21 at 12:01

MattMatt

2,77888 silver badges2121 bronze badges

$endgroup$

add a comment | 

7

$begingroup$

Generally speaking, (particularly if you're new to algebraic topology), a good approach can be we actually write down the maps which give you a homotopy equivalence of spaces. I will give you a few hints to get you going.

HINTS:

• You start by assuming that $(X, x_0) sim (Y, y_0)$ and $(Z, z_0)
  sim (W , w_0)$. This is extremely important. That tells you that you have some maps back and
  forth which satisfy a certain property to do with their composition
  being homotopy equivalent to the identity. Keep these maps, and the homotopies which give you your homotopy equivalences, in mind.




• You can use these maps back and forth to build yourself some maps back and forth on the wedges of spaces $Xvee Z leftrightarrow Yvee W$.




• You then want to show that the composition of these maps is homotopic to the identity on both spaces - the definition of the spaces being homotopy equivalent. To do that, you're going to want to use the homotopies from earlier, which give you that your spaces are homotopy equivalent.

The above is the bare bones of what I imagine is the argument you'll want to run. Give it a shot, and then if you want more details please comment and I can try to say a little more.

share|cite|improve this answer

answered Jun 21 at 12:01

MattMatt

2,77888 silver badges2121 bronze badges

$endgroup$

add a comment | 

$begingroup$

Generally speaking, (particularly if you're new to algebraic topology), a good approach can be we actually write down the maps which give you a homotopy equivalence of spaces. I will give you a few hints to get you going.

HINTS:

• You start by assuming that $(X, x_0) sim (Y, y_0)$ and $(Z, z_0)
  sim (W , w_0)$. This is extremely important. That tells you that you have some maps back and
  forth which satisfy a certain property to do with their composition
  being homotopy equivalent to the identity. Keep these maps, and the homotopies which give you your homotopy equivalences, in mind.




• You can use these maps back and forth to build yourself some maps back and forth on the wedges of spaces $Xvee Z leftrightarrow Yvee W$.




• You then want to show that the composition of these maps is homotopic to the identity on both spaces - the definition of the spaces being homotopy equivalent. To do that, you're going to want to use the homotopies from earlier, which give you that your spaces are homotopy equivalent.

The above is the bare bones of what I imagine is the argument you'll want to run. Give it a shot, and then if you want more details please comment and I can try to say a little more.

share|cite|improve this answer

answered Jun 21 at 12:01

MattMatt

2,77888 silver badges2121 bronze badges

$endgroup$

share|cite|improve this answer

answered Jun 21 at 12:01

MattMatt

2,77888 silver badges2121 bronze badges

answered Jun 21 at 12:01

MattMatt

2,77888 silver badges2121 bronze badges

answered Jun 21 at 12:01

MattMatt

2,77888 silver badges2121 bronze badges