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Hatcher Exercise 3.2.16

1

$begingroup$

Hatcehr Exercise 3.2.16.

Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;mathbb Z)$ and $H^∗(Y;mathbb Z)$
contain no elements of order a power of a given prime $p$, then the same is true for $X×Y$ . [Apply Theorem 3.15 with coefficients in various fields.]

The hint says we can apply

Theorem 3.15. The cross product $H^∗(X;R)⊗_R H^∗(Y;R)→H^∗(X×Y;R)$ is an isomorphism of rings if $X$ and $Y$ are CW complexes and $H^k(Y;R)$ is a finitely generated free $R$-module for all $k$.

Take an element $ain H^∗(X×Y)$ with order $p^n$. I need to produce some elements with orders powers of $p$ in $H^∗(X;mathbb Z)$ and $H^∗(Y;mathbb Z)$.

How can I use his hint? I am thinking about taking $R=mathbb Z_p$ or $operatornameGF(p^n)$ but I don't see the connection to the right solution.

algebraic-topology homology-cohomology cw-complexes

share|cite|improve this question

edited Mar 21 at 9:53

Andrews

1,2812422

asked Mar 19 at 21:26

No OneNo One

2,1051519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  By the theorem (with $R=Bbb Z$), we can identify $a$ with $sum_ia_iotimes b_i$..
  $endgroup$
  – Berci
  Mar 19 at 21:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you need more help, if you know all the orders of the $a_i otimes b_i$ you can compute what the order of $Sigma_i a_i otimes b_i$ has to divide by Lagrange's theorem.
  $endgroup$
  – Connor Malin
  Mar 20 at 0:59
  

  
  
  
  

add a comment | 

1

$begingroup$

Hatcehr Exercise 3.2.16.

Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;mathbb Z)$ and $H^∗(Y;mathbb Z)$
contain no elements of order a power of a given prime $p$, then the same is true for $X×Y$ . [Apply Theorem 3.15 with coefficients in various fields.]

The hint says we can apply

Theorem 3.15. The cross product $H^∗(X;R)⊗_R H^∗(Y;R)→H^∗(X×Y;R)$ is an isomorphism of rings if $X$ and $Y$ are CW complexes and $H^k(Y;R)$ is a finitely generated free $R$-module for all $k$.

Take an element $ain H^∗(X×Y)$ with order $p^n$. I need to produce some elements with orders powers of $p$ in $H^∗(X;mathbb Z)$ and $H^∗(Y;mathbb Z)$.

How can I use his hint? I am thinking about taking $R=mathbb Z_p$ or $operatornameGF(p^n)$ but I don't see the connection to the right solution.

algebraic-topology homology-cohomology cw-complexes

share|cite|improve this question

edited Mar 21 at 9:53

Andrews

1,2812422

asked Mar 19 at 21:26

No OneNo One

2,1051519

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  By the theorem (with $R=Bbb Z$), we can identify $a$ with $sum_ia_iotimes b_i$..
  $endgroup$
  – Berci
  Mar 19 at 21:36
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  If you need more help, if you know all the orders of the $a_i otimes b_i$ you can compute what the order of $Sigma_i a_i otimes b_i$ has to divide by Lagrange's theorem.
  $endgroup$
  – Connor Malin
  Mar 20 at 0:59
  

  
  
  
  

add a comment | 

$begingroup$

Hatcehr Exercise 3.2.16.

Show that if $X$ and $Y$ are finite CW complexes such that $H^∗(X;mathbb Z)$ and $H^∗(Y;mathbb Z)$
contain no elements of order a power of a given prime $p$, then the same is true for $X×Y$ . [Apply Theorem 3.15 with coefficients in various fields.]

The hint says we can apply

Theorem 3.15. The cross product $H^∗(X;R)⊗_R H^∗(Y;R)→H^∗(X×Y;R)$ is an isomorphism of rings if $X$ and $Y$ are CW complexes and $H^k(Y;R)$ is a finitely generated free $R$-module for all $k$.

Take an element $ain H^∗(X×Y)$ with order $p^n$. I need to produce some elements with orders powers of $p$ in $H^∗(X;mathbb Z)$ and $H^∗(Y;mathbb Z)$.

How can I use his hint? I am thinking about taking $R=mathbb Z_p$ or $operatornameGF(p^n)$ but I don't see the connection to the right solution.

algebraic-topology homology-cohomology cw-complexes

share|cite|improve this question

edited Mar 21 at 9:53

Andrews

1,2812422

asked Mar 19 at 21:26

No OneNo One

2,1051519

$endgroup$

share|cite|improve this question

edited Mar 21 at 9:53

Andrews

1,2812422

asked Mar 19 at 21:26

No OneNo One

2,1051519

share|cite|improve this question

edited Mar 21 at 9:53

Andrews

1,2812422

asked Mar 19 at 21:26

No OneNo One

2,1051519

edited Mar 21 at 9:53

Andrews

1,2812422

edited Mar 21 at 9:53

Andrews

1,2812422

asked Mar 19 at 21:26

No OneNo One

2,1051519

asked Mar 19 at 21:26

No OneNo One

2,1051519