Computing cohomology of dihedral group in detail











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So I tried to compute the cohomology of $D_{2n}$, for n odd , $H^{k}(D_{2n}, Bbb Z)$. using Lyndon SS. I have obtained a few obstacles:





My computation, using the fact that there is a $C_2$ action $H^q(C_m, Bbb Z)$, shows that my $E_2^{pq}$ page looks like this:



$$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
$$ cdots 0 cdots $$ $$ cdots 0 cdots $$ $$ cdots 0 cdots $$
$$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
$$ cdots 0 cdots $$
$$ cdots 0 cdots $$
$$ cdots 0 cdots $$
$$ Bbb Z quad 0 quad Bbb Z/2 quad 0 quad Bbb Z/2 quad 0 quad cdots $$



Is this correct?





I suspect the correct $E_2$ page is more or less similar. But I am still unclear how this gives $H^k(D_{2n},Bbb Z)$.



a) How is this $E_2$ page related to $H^k(D_{2n},Bbb Z)$? I only know the case when page 2 collapses to an axis.



b) I suppose one has to show all differential are $0$ to compute $E_infty^{p,q}$. In my case this is simple, since all are $0$.



c) Even if we know the $E_infty$ page does this give $H_n$ - surely we cannot just take direct sum? **





** Supposing my calucations were correct - then we can go to c) directly: What we have here is the relation when $n equiv 0 pmod 4$
$$ 0 rightarrow C_2 rightarrow H^n rightarrow C_m rightarrow 0 $$










share|cite|improve this question




























    up vote
    3
    down vote

    favorite












    So I tried to compute the cohomology of $D_{2n}$, for n odd , $H^{k}(D_{2n}, Bbb Z)$. using Lyndon SS. I have obtained a few obstacles:





    My computation, using the fact that there is a $C_2$ action $H^q(C_m, Bbb Z)$, shows that my $E_2^{pq}$ page looks like this:



    $$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
    $$ cdots 0 cdots $$ $$ cdots 0 cdots $$ $$ cdots 0 cdots $$
    $$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
    $$ cdots 0 cdots $$
    $$ cdots 0 cdots $$
    $$ cdots 0 cdots $$
    $$ Bbb Z quad 0 quad Bbb Z/2 quad 0 quad Bbb Z/2 quad 0 quad cdots $$



    Is this correct?





    I suspect the correct $E_2$ page is more or less similar. But I am still unclear how this gives $H^k(D_{2n},Bbb Z)$.



    a) How is this $E_2$ page related to $H^k(D_{2n},Bbb Z)$? I only know the case when page 2 collapses to an axis.



    b) I suppose one has to show all differential are $0$ to compute $E_infty^{p,q}$. In my case this is simple, since all are $0$.



    c) Even if we know the $E_infty$ page does this give $H_n$ - surely we cannot just take direct sum? **





    ** Supposing my calucations were correct - then we can go to c) directly: What we have here is the relation when $n equiv 0 pmod 4$
    $$ 0 rightarrow C_2 rightarrow H^n rightarrow C_m rightarrow 0 $$










    share|cite|improve this question


























      up vote
      3
      down vote

      favorite









      up vote
      3
      down vote

      favorite











      So I tried to compute the cohomology of $D_{2n}$, for n odd , $H^{k}(D_{2n}, Bbb Z)$. using Lyndon SS. I have obtained a few obstacles:





      My computation, using the fact that there is a $C_2$ action $H^q(C_m, Bbb Z)$, shows that my $E_2^{pq}$ page looks like this:



      $$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
      $$ cdots 0 cdots $$ $$ cdots 0 cdots $$ $$ cdots 0 cdots $$
      $$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
      $$ cdots 0 cdots $$
      $$ cdots 0 cdots $$
      $$ cdots 0 cdots $$
      $$ Bbb Z quad 0 quad Bbb Z/2 quad 0 quad Bbb Z/2 quad 0 quad cdots $$



      Is this correct?





      I suspect the correct $E_2$ page is more or less similar. But I am still unclear how this gives $H^k(D_{2n},Bbb Z)$.



      a) How is this $E_2$ page related to $H^k(D_{2n},Bbb Z)$? I only know the case when page 2 collapses to an axis.



      b) I suppose one has to show all differential are $0$ to compute $E_infty^{p,q}$. In my case this is simple, since all are $0$.



      c) Even if we know the $E_infty$ page does this give $H_n$ - surely we cannot just take direct sum? **





      ** Supposing my calucations were correct - then we can go to c) directly: What we have here is the relation when $n equiv 0 pmod 4$
      $$ 0 rightarrow C_2 rightarrow H^n rightarrow C_m rightarrow 0 $$










      share|cite|improve this question















      So I tried to compute the cohomology of $D_{2n}$, for n odd , $H^{k}(D_{2n}, Bbb Z)$. using Lyndon SS. I have obtained a few obstacles:





      My computation, using the fact that there is a $C_2$ action $H^q(C_m, Bbb Z)$, shows that my $E_2^{pq}$ page looks like this:



      $$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
      $$ cdots 0 cdots $$ $$ cdots 0 cdots $$ $$ cdots 0 cdots $$
      $$ Bbb Z/m quad 0 quad 0 quad 0 quad 0 quad 0 quad cdots $$
      $$ cdots 0 cdots $$
      $$ cdots 0 cdots $$
      $$ cdots 0 cdots $$
      $$ Bbb Z quad 0 quad Bbb Z/2 quad 0 quad Bbb Z/2 quad 0 quad cdots $$



      Is this correct?





      I suspect the correct $E_2$ page is more or less similar. But I am still unclear how this gives $H^k(D_{2n},Bbb Z)$.



      a) How is this $E_2$ page related to $H^k(D_{2n},Bbb Z)$? I only know the case when page 2 collapses to an axis.



      b) I suppose one has to show all differential are $0$ to compute $E_infty^{p,q}$. In my case this is simple, since all are $0$.



      c) Even if we know the $E_infty$ page does this give $H_n$ - surely we cannot just take direct sum? **





      ** Supposing my calucations were correct - then we can go to c) directly: What we have here is the relation when $n equiv 0 pmod 4$
      $$ 0 rightarrow C_2 rightarrow H^n rightarrow C_m rightarrow 0 $$







      homology-cohomology homological-algebra group-cohomology






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




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      edited 23 hours ago

























      asked yesterday









      CL.

      2,0102822




      2,0102822






















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          Your $E_2$ page is correct.



          To continue, observe that everything nonzero in your $E_2$ page lies in some even bidegree $(2p,2q)$, and that the grading of the differential $d^i$ on the $E_i$ page is of bidegree $(1-i, i)$, and in particular, changes the parity of at least one of the degrees; in particular, $d^i$ must be identically zero for all $i$. So $E_2 = E_infty$. This is always true for spectral sequences whose elements are concentrated in bidegrees $(p,q)$ with $p+q$ even.



          Next, let us recall what it means that this spectral sequence converges: the calculational part of that means that the homology groups $H^k(D_{2n}, Bbb Z)$ have a filtration $0 = F^0 supset cdots supset F^i supset F^{i+1} supset cdots$ and the associated graded $$text{gr}^p H^{p+q}(D_{2n}, Bbb Z) = E^{p,q}_infty.$$



          For us, on the lines with $p + q = 4n+2$, we see that there is only one nontrivial subquotient $F^p/F^{p+1}$, and in particular, $H^{4n+2} = E^infty_{4n+2,0}$.



          On the lines with $p + q = 4n >0$, we have two terms. There is $E^infty_{4n,0} = F^0/F^1 H^{4n}$; because there is nothing until $q = 4n$, we see that $F^1 = F^2 = cdots = F^{4n}$, but that $F^{4n+1} = 0$, and thus $F^{4n} = E^infty_{0,4n}.$



          In particular, we have a short exact sequence $0 to E^infty_{0,4n} to H^{4n} to E^infty_{4n,0} to 0$. Resolving this is what is usually called "solving a filtration problem". If there are more than two nonzero terms on the $p+q = n$ line, then one has to resolve these iteratively, starting from $E^infty_{0,4n}$ and walking down the line.



          In this particular case, $E^infty_{4n,0} = Bbb Z/2$, and $E^infty_{0,4n} = Bbb Z/m$ where $m$ is odd. Extensions of abelian groups where the subgroup and quotient are of coprime order always split, and so in particular $H^{4n} = Bbb Z/2 oplus Bbb Z/m cong Bbb Z/2m$ in this case.



          If you are also interested in the multiplicative structure, one can write this ring as $Bbb Z[c_1, c_2]/(2c_1, c_1^2 = mc_2)$, following the strategy of this answer for $S_3 = D_6$; there is no essential difference between that case and $D_{4n+2}$ in general.






          share|cite|improve this answer























          • I don't want to bump again, but in the final paragraph $|c_1| = 2$ and $|c_2| = 4$; that is, $c_1$ is the nonzero element of $H^2$ and $c_2$ is a choice of generator for $H^4 = Bbb Z/2m$.
            – Mike Miller
            11 hours ago










          • Thanks a lot. Just curious, is there a reason why your notation went from $E_infty $ to $E^infty$?
            – CL.
            49 mins ago













          Your Answer





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          up vote
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          accepted










          Your $E_2$ page is correct.



          To continue, observe that everything nonzero in your $E_2$ page lies in some even bidegree $(2p,2q)$, and that the grading of the differential $d^i$ on the $E_i$ page is of bidegree $(1-i, i)$, and in particular, changes the parity of at least one of the degrees; in particular, $d^i$ must be identically zero for all $i$. So $E_2 = E_infty$. This is always true for spectral sequences whose elements are concentrated in bidegrees $(p,q)$ with $p+q$ even.



          Next, let us recall what it means that this spectral sequence converges: the calculational part of that means that the homology groups $H^k(D_{2n}, Bbb Z)$ have a filtration $0 = F^0 supset cdots supset F^i supset F^{i+1} supset cdots$ and the associated graded $$text{gr}^p H^{p+q}(D_{2n}, Bbb Z) = E^{p,q}_infty.$$



          For us, on the lines with $p + q = 4n+2$, we see that there is only one nontrivial subquotient $F^p/F^{p+1}$, and in particular, $H^{4n+2} = E^infty_{4n+2,0}$.



          On the lines with $p + q = 4n >0$, we have two terms. There is $E^infty_{4n,0} = F^0/F^1 H^{4n}$; because there is nothing until $q = 4n$, we see that $F^1 = F^2 = cdots = F^{4n}$, but that $F^{4n+1} = 0$, and thus $F^{4n} = E^infty_{0,4n}.$



          In particular, we have a short exact sequence $0 to E^infty_{0,4n} to H^{4n} to E^infty_{4n,0} to 0$. Resolving this is what is usually called "solving a filtration problem". If there are more than two nonzero terms on the $p+q = n$ line, then one has to resolve these iteratively, starting from $E^infty_{0,4n}$ and walking down the line.



          In this particular case, $E^infty_{4n,0} = Bbb Z/2$, and $E^infty_{0,4n} = Bbb Z/m$ where $m$ is odd. Extensions of abelian groups where the subgroup and quotient are of coprime order always split, and so in particular $H^{4n} = Bbb Z/2 oplus Bbb Z/m cong Bbb Z/2m$ in this case.



          If you are also interested in the multiplicative structure, one can write this ring as $Bbb Z[c_1, c_2]/(2c_1, c_1^2 = mc_2)$, following the strategy of this answer for $S_3 = D_6$; there is no essential difference between that case and $D_{4n+2}$ in general.






          share|cite|improve this answer























          • I don't want to bump again, but in the final paragraph $|c_1| = 2$ and $|c_2| = 4$; that is, $c_1$ is the nonzero element of $H^2$ and $c_2$ is a choice of generator for $H^4 = Bbb Z/2m$.
            – Mike Miller
            11 hours ago










          • Thanks a lot. Just curious, is there a reason why your notation went from $E_infty $ to $E^infty$?
            – CL.
            49 mins ago

















          up vote
          2
          down vote



          accepted










          Your $E_2$ page is correct.



          To continue, observe that everything nonzero in your $E_2$ page lies in some even bidegree $(2p,2q)$, and that the grading of the differential $d^i$ on the $E_i$ page is of bidegree $(1-i, i)$, and in particular, changes the parity of at least one of the degrees; in particular, $d^i$ must be identically zero for all $i$. So $E_2 = E_infty$. This is always true for spectral sequences whose elements are concentrated in bidegrees $(p,q)$ with $p+q$ even.



          Next, let us recall what it means that this spectral sequence converges: the calculational part of that means that the homology groups $H^k(D_{2n}, Bbb Z)$ have a filtration $0 = F^0 supset cdots supset F^i supset F^{i+1} supset cdots$ and the associated graded $$text{gr}^p H^{p+q}(D_{2n}, Bbb Z) = E^{p,q}_infty.$$



          For us, on the lines with $p + q = 4n+2$, we see that there is only one nontrivial subquotient $F^p/F^{p+1}$, and in particular, $H^{4n+2} = E^infty_{4n+2,0}$.



          On the lines with $p + q = 4n >0$, we have two terms. There is $E^infty_{4n,0} = F^0/F^1 H^{4n}$; because there is nothing until $q = 4n$, we see that $F^1 = F^2 = cdots = F^{4n}$, but that $F^{4n+1} = 0$, and thus $F^{4n} = E^infty_{0,4n}.$



          In particular, we have a short exact sequence $0 to E^infty_{0,4n} to H^{4n} to E^infty_{4n,0} to 0$. Resolving this is what is usually called "solving a filtration problem". If there are more than two nonzero terms on the $p+q = n$ line, then one has to resolve these iteratively, starting from $E^infty_{0,4n}$ and walking down the line.



          In this particular case, $E^infty_{4n,0} = Bbb Z/2$, and $E^infty_{0,4n} = Bbb Z/m$ where $m$ is odd. Extensions of abelian groups where the subgroup and quotient are of coprime order always split, and so in particular $H^{4n} = Bbb Z/2 oplus Bbb Z/m cong Bbb Z/2m$ in this case.



          If you are also interested in the multiplicative structure, one can write this ring as $Bbb Z[c_1, c_2]/(2c_1, c_1^2 = mc_2)$, following the strategy of this answer for $S_3 = D_6$; there is no essential difference between that case and $D_{4n+2}$ in general.






          share|cite|improve this answer























          • I don't want to bump again, but in the final paragraph $|c_1| = 2$ and $|c_2| = 4$; that is, $c_1$ is the nonzero element of $H^2$ and $c_2$ is a choice of generator for $H^4 = Bbb Z/2m$.
            – Mike Miller
            11 hours ago










          • Thanks a lot. Just curious, is there a reason why your notation went from $E_infty $ to $E^infty$?
            – CL.
            49 mins ago















          up vote
          2
          down vote



          accepted







          up vote
          2
          down vote



          accepted






          Your $E_2$ page is correct.



          To continue, observe that everything nonzero in your $E_2$ page lies in some even bidegree $(2p,2q)$, and that the grading of the differential $d^i$ on the $E_i$ page is of bidegree $(1-i, i)$, and in particular, changes the parity of at least one of the degrees; in particular, $d^i$ must be identically zero for all $i$. So $E_2 = E_infty$. This is always true for spectral sequences whose elements are concentrated in bidegrees $(p,q)$ with $p+q$ even.



          Next, let us recall what it means that this spectral sequence converges: the calculational part of that means that the homology groups $H^k(D_{2n}, Bbb Z)$ have a filtration $0 = F^0 supset cdots supset F^i supset F^{i+1} supset cdots$ and the associated graded $$text{gr}^p H^{p+q}(D_{2n}, Bbb Z) = E^{p,q}_infty.$$



          For us, on the lines with $p + q = 4n+2$, we see that there is only one nontrivial subquotient $F^p/F^{p+1}$, and in particular, $H^{4n+2} = E^infty_{4n+2,0}$.



          On the lines with $p + q = 4n >0$, we have two terms. There is $E^infty_{4n,0} = F^0/F^1 H^{4n}$; because there is nothing until $q = 4n$, we see that $F^1 = F^2 = cdots = F^{4n}$, but that $F^{4n+1} = 0$, and thus $F^{4n} = E^infty_{0,4n}.$



          In particular, we have a short exact sequence $0 to E^infty_{0,4n} to H^{4n} to E^infty_{4n,0} to 0$. Resolving this is what is usually called "solving a filtration problem". If there are more than two nonzero terms on the $p+q = n$ line, then one has to resolve these iteratively, starting from $E^infty_{0,4n}$ and walking down the line.



          In this particular case, $E^infty_{4n,0} = Bbb Z/2$, and $E^infty_{0,4n} = Bbb Z/m$ where $m$ is odd. Extensions of abelian groups where the subgroup and quotient are of coprime order always split, and so in particular $H^{4n} = Bbb Z/2 oplus Bbb Z/m cong Bbb Z/2m$ in this case.



          If you are also interested in the multiplicative structure, one can write this ring as $Bbb Z[c_1, c_2]/(2c_1, c_1^2 = mc_2)$, following the strategy of this answer for $S_3 = D_6$; there is no essential difference between that case and $D_{4n+2}$ in general.






          share|cite|improve this answer














          Your $E_2$ page is correct.



          To continue, observe that everything nonzero in your $E_2$ page lies in some even bidegree $(2p,2q)$, and that the grading of the differential $d^i$ on the $E_i$ page is of bidegree $(1-i, i)$, and in particular, changes the parity of at least one of the degrees; in particular, $d^i$ must be identically zero for all $i$. So $E_2 = E_infty$. This is always true for spectral sequences whose elements are concentrated in bidegrees $(p,q)$ with $p+q$ even.



          Next, let us recall what it means that this spectral sequence converges: the calculational part of that means that the homology groups $H^k(D_{2n}, Bbb Z)$ have a filtration $0 = F^0 supset cdots supset F^i supset F^{i+1} supset cdots$ and the associated graded $$text{gr}^p H^{p+q}(D_{2n}, Bbb Z) = E^{p,q}_infty.$$



          For us, on the lines with $p + q = 4n+2$, we see that there is only one nontrivial subquotient $F^p/F^{p+1}$, and in particular, $H^{4n+2} = E^infty_{4n+2,0}$.



          On the lines with $p + q = 4n >0$, we have two terms. There is $E^infty_{4n,0} = F^0/F^1 H^{4n}$; because there is nothing until $q = 4n$, we see that $F^1 = F^2 = cdots = F^{4n}$, but that $F^{4n+1} = 0$, and thus $F^{4n} = E^infty_{0,4n}.$



          In particular, we have a short exact sequence $0 to E^infty_{0,4n} to H^{4n} to E^infty_{4n,0} to 0$. Resolving this is what is usually called "solving a filtration problem". If there are more than two nonzero terms on the $p+q = n$ line, then one has to resolve these iteratively, starting from $E^infty_{0,4n}$ and walking down the line.



          In this particular case, $E^infty_{4n,0} = Bbb Z/2$, and $E^infty_{0,4n} = Bbb Z/m$ where $m$ is odd. Extensions of abelian groups where the subgroup and quotient are of coprime order always split, and so in particular $H^{4n} = Bbb Z/2 oplus Bbb Z/m cong Bbb Z/2m$ in this case.



          If you are also interested in the multiplicative structure, one can write this ring as $Bbb Z[c_1, c_2]/(2c_1, c_1^2 = mc_2)$, following the strategy of this answer for $S_3 = D_6$; there is no essential difference between that case and $D_{4n+2}$ in general.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 20 hours ago

























          answered 22 hours ago









          Mike Miller

          35.1k467131




          35.1k467131












          • I don't want to bump again, but in the final paragraph $|c_1| = 2$ and $|c_2| = 4$; that is, $c_1$ is the nonzero element of $H^2$ and $c_2$ is a choice of generator for $H^4 = Bbb Z/2m$.
            – Mike Miller
            11 hours ago










          • Thanks a lot. Just curious, is there a reason why your notation went from $E_infty $ to $E^infty$?
            – CL.
            49 mins ago




















          • I don't want to bump again, but in the final paragraph $|c_1| = 2$ and $|c_2| = 4$; that is, $c_1$ is the nonzero element of $H^2$ and $c_2$ is a choice of generator for $H^4 = Bbb Z/2m$.
            – Mike Miller
            11 hours ago










          • Thanks a lot. Just curious, is there a reason why your notation went from $E_infty $ to $E^infty$?
            – CL.
            49 mins ago


















          I don't want to bump again, but in the final paragraph $|c_1| = 2$ and $|c_2| = 4$; that is, $c_1$ is the nonzero element of $H^2$ and $c_2$ is a choice of generator for $H^4 = Bbb Z/2m$.
          – Mike Miller
          11 hours ago




          I don't want to bump again, but in the final paragraph $|c_1| = 2$ and $|c_2| = 4$; that is, $c_1$ is the nonzero element of $H^2$ and $c_2$ is a choice of generator for $H^4 = Bbb Z/2m$.
          – Mike Miller
          11 hours ago












          Thanks a lot. Just curious, is there a reason why your notation went from $E_infty $ to $E^infty$?
          – CL.
          49 mins ago






          Thanks a lot. Just curious, is there a reason why your notation went from $E_infty $ to $E^infty$?
          – CL.
          49 mins ago




















           

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