# filtered complexes and spectral sequences from simplicial complexes

To make a filtered complex from a list of simplicial complexes we first need to make some simplicial complexes.

 i1 : R = QQ[x,y,z,w]; i2 : a = simplicialComplex {x*y*z, x*y, y*z, w*z} o2 = | zw xyz | o2 : SimplicialComplex i3 : b = simplicialComplex {x*y, w} o3 = | w xy | o3 : SimplicialComplex i4 : c = simplicialComplex {x,w} o4 = | w x | o4 : SimplicialComplex

Note that $b$ is a simplicial subcomplex of $a$ and that $c$ is a simplicial subcomplex of $b$. Let's now create a filtered complex.

 i5 : K = filteredComplex{a,b,c} o5 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 -1 0 1 2 0 : image | 1 | <-- image | 1 0 | <-- image 0 <-- image 0 | 0 0 | -1 | 0 0 | 1 2 | 0 1 | 0 1 : image | 1 | <-- image | 1 0 0 | <-- image | 1 | <-- image 0 | 0 1 0 | | 0 | -1 | 0 0 0 | | 0 | 2 | 0 0 1 | | 0 | 0 1 1 4 4 1 2 : QQ <-- QQ <-- QQ <-- QQ -1 0 1 2 o5 : FilteredComplex

The associated spectral sequence takes the form:

 i6 : E = spectralSequence K o6 = E o6 : SpectralSequence

Let's view some pages and maps of these pages.

 i7 : E^0 +-----------------------------+----------------------------------------+---------------------+ | | | 1 | o7 = |image | 1 0 0 0 0 0 0 0 0 0 ||image | 1 | |QQ | | | 0 0 0 0 0 0 0 0 0 0 || | 0 | | | | | 0 0 0 0 0 0 0 0 0 0 || | 0 | |{2, 0} | | | 0 1 0 0 0 0 0 0 0 0 || | 0 | | | | | | | |{0, 0} |{1, 0} | | +-----------------------------+----------------------------------------+---------------------+ |image | 1 0 0 0 0 0 | |subquotient (| 1 0 0 1 0 |, | 1 0 |) |cokernel | 1 | | | | | 0 1 0 0 0 | | 0 0 | | | 0 | | |{0, -1} | | 0 0 0 0 0 | | 0 0 | | | 0 | | | | | 0 0 1 0 1 | | 0 1 | | | 0 | | | | | | | |{1, -1} |{2, -1} | +-----------------------------+----------------------------------------+---------------------+ |0 |subquotient (| 1 -1 -1 1 |, | -1 -1 1 |)|cokernel | 1 1 0 0 || | | | | -1 0 1 0 || |{0, -2} |{1, -2} | | 0 0 0 0 || | | | | 0 0 0 1 || | | | | | | |{2, -2} | +-----------------------------+----------------------------------------+---------------------+ o7 : SpectralSequencePage i8 : F0 = minimalPresentation(E^0) +-------+-------+-------+ | 2 | 1 | 1 | o8 = |QQ |QQ |QQ | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | 1 | 1 | 3 | |QQ |QQ |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -2}|{1, -2}|{2, -2}| +-------+-------+-------+ o8 : SpectralSequencePage i9 : E^0 .dd o9 = {-1, 0} : image 0 <----- image 0 : {-1, 1} 0 {-1, 1} : image 0 <----- image 0 : {-1, 2} 0 {-1, 2} : image 0 <----- image 0 : {-1, 3} 0 {2, -4} : 0 <----- cokernel | -1 -1 -1 1 | : {2, -3} 0 {2, -3} : cokernel | -1 -1 -1 1 | <----- cokernel | 1 1 0 0 | : {2, -2} 0 | -1 0 1 0 | | 0 0 0 0 | | 0 0 0 1 | {2, -2} : cokernel | 1 1 0 0 | <----------------- cokernel | 1 | : {2, -1} | -1 0 1 0 | | 0 0 0 0 | | 0 | | 0 0 0 0 | | 0 0 0 0 | | 0 | | 0 0 0 1 | | 0 -1 -1 1 | | 0 | | 0 0 0 0 | 1 {2, -1} : cokernel | 1 | <---------- QQ : {2, 0} | 0 | | 0 | | 0 | | 1 | | 0 | | -1 | | 0 | {1, -3} : 0 <----- subquotient (| 1 -1 -1 1 |, | -1 -1 1 |) : {1, -2} 0 {1, -2} : subquotient (| 1 -1 -1 1 |, | -1 -1 1 |) <----- subquotient (| 1 0 0 1 0 |, | 1 0 |) : {1, -1} 0 | 0 1 0 0 0 | | 0 0 | | 0 0 0 0 0 | | 0 0 | | 0 0 1 0 1 | | 0 1 | {1, -1} : subquotient (| 1 0 0 1 0 |, | 1 0 |) <---------- image | 1 | : {1, 0} | 0 1 0 0 0 | | 0 0 | | 0 | | 0 | | 0 0 0 0 0 | | 0 0 | | -1 | | 0 | | 0 0 1 0 1 | | 0 1 | | 0 | | 0 | | 0 | | 0 | {1, 0} : image | 1 | <----- image 0 : {1, 1} | 0 | 0 | 0 | | 0 | {0, -2} : 0 <----- image | 1 0 0 0 0 0 | : {0, -1} 0 {0, -1} : image | 1 0 0 0 0 0 | <----------------------------- image | 1 0 0 0 0 0 0 0 0 0 | : {0, 0} | -1 -1 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 1 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | | 0 0 0 0 0 0 0 0 0 0 | {0, 0} : image | 1 0 0 0 0 0 0 0 0 0 | <----- image 0 : {0, 1} | 0 0 0 0 0 0 0 0 0 0 | 0 | 0 0 0 0 0 0 0 0 0 0 | | 0 1 0 0 0 0 0 0 0 0 | {0, 1} : image 0 <----- image 0 : {0, 2} 0 {-1, -1} : 0 <----- image 0 : {-1, 0} 0 o9 : SpectralSequencePageMap i10 : F0.dd o10 = {-1, 0} : 0 <----- 0 : {-1, 1} 0 {-1, 1} : 0 <----- 0 : {-1, 2} 0 {-1, 2} : 0 <----- 0 : {-1, 3} 0 {2, -4} : 0 <----- 0 : {2, -3} 0 1 {2, -3} : 0 <----- QQ : {2, -2} 0 1 3 {2, -2} : QQ <--------------- QQ : {2, -1} | -1 -1 1 | 3 1 {2, -1} : QQ <---------- QQ : {2, 0} | 1 | | -1 | | 0 | {1, -3} : 0 <----- 0 : {1, -2} 0 1 {1, -2} : 0 <----- QQ : {1, -1} 0 1 1 {1, -1} : QQ <---------- QQ : {1, 0} | -1 | 1 {1, 0} : QQ <----- 0 : {1, 1} 0 1 {0, -2} : 0 <----- QQ : {0, -1} 0 1 2 {0, -1} : QQ <------------- QQ : {0, 0} | -1 -1 | 2 {0, 0} : QQ <----- 0 : {0, 1} 0 {0, 1} : 0 <----- 0 : {0, 2} 0 {-1, -1} : 0 <----- 0 : {-1, 0} 0 o10 : SpectralSequencePageMap i11 : E^1 +----------------------------------------+------------------------------------------+---------------------------------------+ o11 = |image | -1 0 0 0 0 | |image 0 |image 0 | | | 0 0 0 0 0 | | | | | | 0 0 0 0 0 | |{1, 0} |{2, 0} | | | 1 0 0 0 0 | | | | | | | | |{0, 0} | | | +----------------------------------------+------------------------------------------+---------------------------------------+ |subquotient (| 1 -1 -1 0 |, | -1 -1 0 |)|subquotient (| 1 0 0 1 1 0 |, | 1 1 0 |)|subquotient (| 1 0 0 -1 1 |, | -1 1 |)| | | | 0 1 0 -1 0 0 | | -1 0 0 | | | 0 -1 1 1 0 | | 1 0 | | |{0, -1} | | 0 0 0 0 0 0 | | 0 0 0 | | | 0 1 0 -1 0 | | -1 0 | | | | | 0 0 1 0 0 1 | | 0 0 1 | | | 0 0 1 0 0 | | 0 0 | | | | | | | |{1, -1} |{2, -1} | +----------------------------------------+------------------------------------------+---------------------------------------+ o11 : SpectralSequencePage i12 : F1 = minimalPresentation(E^1) +-------+-------+-------+ | 1 | | | o12 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o12 : SpectralSequencePage i13 : E^1 .dd o13 = {-2, 1} : image 0 <----- image 0 : {-1, 1} 0 {-2, 2} : image 0 <----- image 0 : {-1, 2} 0 {-2, 3} : image 0 <----- image 0 : {-1, 3} 0 {1, -3} : 0 <----- cokernel | -1 -1 -1 -1 1 | : {2, -3} 0 {1, -2} : subquotient (| 1 -1 -1 -1 1 |, | -1 -1 -1 1 |) <----- cokernel | 1 1 0 0 1 0 0 | : {2, -2} 0 | -1 0 1 0 0 1 0 | | 0 -1 -1 1 0 0 0 | | 0 0 0 -1 0 0 1 | {1, -1} : subquotient (| 1 0 0 1 1 0 |, | 1 1 0 |) <----- subquotient (| 1 0 0 -1 1 |, | -1 1 |) : {2, -1} | 0 1 0 -1 0 0 | | -1 0 0 | 0 | 0 -1 1 1 0 | | 1 0 | | 0 0 0 0 0 0 | | 0 0 0 | | 0 1 0 -1 0 | | -1 0 | | 0 0 1 0 0 1 | | 0 0 1 | | 0 0 1 0 0 | | 0 0 | {1, 0} : image 0 <----- image 0 : {2, 0} 0 {0, -2} : 0 <----- subquotient (| 1 -1 -1 -1 1 |, | -1 -1 -1 1 |) : {1, -2} 0 {0, -1} : subquotient (| 1 -1 -1 0 |, | -1 -1 0 |) <----- subquotient (| 1 0 0 1 1 0 |, | 1 1 0 |) : {1, -1} 0 | 0 1 0 -1 0 0 | | -1 0 0 | | 0 0 0 0 0 0 | | 0 0 0 | | 0 0 1 0 0 1 | | 0 0 1 | {0, 0} : image | -1 0 0 0 0 | <----- image 0 : {1, 0} | 0 0 0 0 0 | 0 | 0 0 0 0 0 | | 1 0 0 0 0 | {0, 1} : image 0 <----- image 0 : {1, 1} 0 {-1, -1} : 0 <----- subquotient (| 1 -1 -1 0 |, | -1 -1 0 |) : {0, -1} 0 {-1, 0} : image 0 <----- image | -1 0 0 0 0 | : {0, 0} 0 | 0 0 0 0 0 | | 0 0 0 0 0 | | 1 0 0 0 0 | {-1, 1} : image 0 <----- image 0 : {0, 1} 0 {-1, 2} : image 0 <----- image 0 : {0, 2} 0 {-2, 0} : 0 <----- image 0 : {-1, 0} 0 o13 : SpectralSequencePageMap i14 : F1.dd o14 = {-2, 1} : 0 <----- 0 : {-1, 1} 0 {-2, 2} : 0 <----- 0 : {-1, 2} 0 {-2, 3} : 0 <----- 0 : {-1, 3} 0 {1, -3} : 0 <----- 0 : {2, -3} 0 {1, -2} : 0 <----- 0 : {2, -2} 0 1 {1, -1} : 0 <----- QQ : {2, -1} 0 {1, 0} : 0 <----- 0 : {2, 0} 0 {0, -2} : 0 <----- 0 : {1, -2} 0 {0, -1} : 0 <----- 0 : {1, -1} 0 1 {0, 0} : QQ <----- 0 : {1, 0} 0 {0, 1} : 0 <----- 0 : {1, 1} 0 {-1, -1} : 0 <----- 0 : {0, -1} 0 1 {-1, 0} : 0 <----- QQ : {0, 0} 0 {-1, 1} : 0 <----- 0 : {0, 1} 0 {-1, 2} : 0 <----- 0 : {0, 2} 0 {-2, 0} : 0 <----- 0 : {-1, 0} 0 o14 : SpectralSequencePageMap i15 : E^2 +-----------------------------------------------+------------------------------------------------------------+-------------------------------------+ o15 = |subquotient (| -1 1 0 0 0 0 |, | 1 0 0 0 0 |)|subquotient (| -1 |, | -1 |) |image 0 | | | 0 -1 0 0 0 0 | | -1 0 0 0 0 | | | 1 | | 1 | | | | | 0 0 0 0 0 0 | | 0 0 0 0 0 | | | -1 | | -1 | |{2, 0} | | | 1 0 0 0 0 0 | | 0 0 0 0 0 | | | 0 | | 0 | | | | | | | |{0, 0} |{1, 0} | | +-----------------------------------------------+------------------------------------------------------------+-------------------------------------+ |subquotient (| 1 -1 -1 -1 0 |, | -1 -1 -1 0 |) |subquotient (| -1 -1 1 1 0 0 1 0 |, | 1 1 0 0 1 0 |)|subquotient (| 1 0 -1 1 |, | -1 1 |)| | | | 1 0 -1 0 1 0 0 0 | | -1 0 1 0 0 0 | | | -1 1 1 0 | | 1 0 | | |{0, -1} | | 0 0 0 -1 -1 1 0 0 | | 0 -1 -1 1 0 0 | | | 1 0 -1 0 | | -1 0 | | | | | 0 1 0 0 0 -1 0 1 | | 0 0 0 -1 0 1 | | | 0 1 0 0 | | 0 0 | | | | | | | |{1, -1} |{2, -1} | +-----------------------------------------------+------------------------------------------------------------+-------------------------------------+ o15 : SpectralSequencePage i16 : F2 = minimalPresentation(E^2) +-------+-------+-------+ | 1 | | | o16 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o16 : SpectralSequencePage i17 : E^2 .dd o17 = {-3, 2} : image 0 <----- image 0 : {-1, 1} 0 {-3, 3} : image 0 <----- image 0 : {-1, 2} 0 {-3, 4} : image 0 <----- image 0 : {-1, 3} 0 {0, -2} : 0 <----- cokernel | -1 -1 -1 -1 1 | : {2, -3} 0 {0, -1} : subquotient (| 1 -1 -1 -1 0 |, | -1 -1 -1 0 |) <----- cokernel | 1 1 0 0 1 0 0 | : {2, -2} 0 | -1 0 1 0 0 1 0 | | 0 -1 -1 1 0 0 0 | | 0 0 0 -1 0 0 1 | {0, 0} : subquotient (| -1 1 0 0 0 0 |, | 1 0 0 0 0 |) <---------------- subquotient (| 1 0 -1 1 |, | -1 1 |) : {2, -1} | 0 -1 0 0 0 0 | | -1 0 0 0 0 | | 0 -1 0 0 | | -1 1 1 0 | | 1 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 | | 1 0 -1 0 | | -1 0 | | 1 0 0 0 0 0 | | 0 0 0 0 0 | | 0 0 0 0 | | 0 1 0 0 | | 0 0 | | 0 0 0 0 | | 0 0 0 0 | | 0 0 0 0 | {0, 1} : image 0 <----- image 0 : {2, 0} 0 {-1, -1} : 0 <----- subquotient (| 1 -1 -1 -1 -1 1 |, | -1 -1 -1 -1 1 |) : {1, -2} 0 {-1, 0} : subquotient (| 0 -1 -1 0 |, | -1 -1 0 |) <----- subquotient (| -1 -1 1 1 0 0 1 0 |, | 1 1 0 0 1 0 |) : {1, -1} 0 | 1 0 -1 0 1 0 0 0 | | -1 0 1 0 0 0 | | 0 0 0 -1 -1 1 0 0 | | 0 -1 -1 1 0 0 | | 0 1 0 0 0 -1 0 1 | | 0 0 0 -1 0 1 | {-1, 1} : image 0 <----- subquotient (| -1 |, | -1 |) : {1, 0} 0 | 1 | | 1 | | -1 | | -1 | | 0 | | 0 | {-1, 2} : image 0 <----- image 0 : {1, 1} 0 {-2, 0} : 0 <----- subquotient (| 1 -1 -1 -1 0 |, | -1 -1 -1 0 |) : {0, -1} 0 {-2, 1} : image 0 <----- subquotient (| -1 1 0 0 0 0 |, | 1 0 0 0 0 |) : {0, 0} 0 | 0 -1 0 0 0 0 | | -1 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 | | 1 0 0 0 0 0 | | 0 0 0 0 0 | {-2, 2} : image 0 <----- image 0 : {0, 1} 0 {-2, 3} : image 0 <----- image 0 : {0, 2} 0 {-3, 1} : 0 <----- subquotient (| 0 -1 -1 0 |, | -1 -1 0 |) : {-1, 0} 0 o17 : SpectralSequencePageMap i18 : F2.dd o18 = {-3, 2} : 0 <----- 0 : {-1, 1} 0 {-3, 3} : 0 <----- 0 : {-1, 2} 0 {-3, 4} : 0 <----- 0 : {-1, 3} 0 {0, -2} : 0 <----- 0 : {2, -3} 0 {0, -1} : 0 <----- 0 : {2, -2} 0 1 1 {0, 0} : QQ <---------- QQ : {2, -1} | -1 | {0, 1} : 0 <----- 0 : {2, 0} 0 {-1, -1} : 0 <----- 0 : {1, -2} 0 {-1, 0} : 0 <----- 0 : {1, -1} 0 {-1, 1} : 0 <----- 0 : {1, 0} 0 {-1, 2} : 0 <----- 0 : {1, 1} 0 {-2, 0} : 0 <----- 0 : {0, -1} 0 1 {-2, 1} : 0 <----- QQ : {0, 0} 0 {-2, 2} : 0 <----- 0 : {0, 1} 0 {-2, 3} : 0 <----- 0 : {0, 2} 0 {-3, 1} : 0 <----- 0 : {-1, 0} 0 o18 : SpectralSequencePageMap i19 : E^infinity o19 = ++ ++ o19 : Page i20 : (prune E) ^infinity o20 = ++ ++ o20 : Page

If we want the resulting complexes to correspond to the non-reduced homology of the simpicial complexes we set the ReducedHomology option to false.

 i21 : J = filteredComplex({a,b,c}, ReducedHomology => false) o21 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0 -1 0 1 2 0 : image 0 <-- image | 1 0 | <-- image 0 <-- image 0 | 0 0 | -1 | 0 0 | 1 2 | 0 1 | 0 1 : image 0 <-- image | 1 0 0 | <-- image | 1 | <-- image 0 | 0 1 0 | | 0 | -1 | 0 0 0 | | 0 | 2 | 0 0 1 | | 0 | 0 1 4 4 1 2 : image 0 <-- QQ <-- QQ <-- QQ -1 0 1 2 o21 : FilteredComplex

The resulting spectral sequence looks like

 i22 : D = spectralSequence J o22 = D o22 : SpectralSequence i23 : D^0 +-----------------------------+------------------------------------+---------------------+ | | | 1 | o23 = |image | 1 0 0 0 0 0 0 0 0 0 ||image | 1 | |QQ | | | 0 0 0 0 0 0 0 0 0 0 || | 0 | | | | | 0 0 0 0 0 0 0 0 0 0 || | 0 | |{2, 0} | | | 0 1 0 0 0 0 0 0 0 0 || | 0 | | | | | | | |{0, 0} |{1, 0} | | +-----------------------------+------------------------------------+---------------------+ |image 0 |subquotient (| 1 0 0 1 0 |, | 1 0 |)|cokernel | 1 | | | | | 0 1 0 0 0 | | 0 0 | | | 0 | | |{0, -1} | | 0 0 0 0 0 | | 0 0 | | | 0 | | | | | 0 0 1 0 1 | | 0 1 | | | 0 | | | | | | | |{1, -1} |{2, -1} | +-----------------------------+------------------------------------+---------------------+ |0 |image 0 |cokernel | 1 1 0 0 || | | | | -1 0 1 0 || |{0, -2} |{1, -2} | | 0 0 0 0 || | | | | 0 0 0 1 || | | | | | | |{2, -2} | +-----------------------------+------------------------------------+---------------------+ o23 : SpectralSequencePage i24 : G0 = minimalPresentation(D^0) +-------+-------+-------+ | 2 | 1 | 1 | o24 = |QQ |QQ |QQ | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | 1 | 3 | |0 |QQ |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -2}|{1, -2}|{2, -2}| +-------+-------+-------+ o24 : SpectralSequencePage i25 : G0.dd o25 = {-1, 0} : 0 <----- 0 : {-1, 1} 0 {-1, 1} : 0 <----- 0 : {-1, 2} 0 {-1, 2} : 0 <----- 0 : {-1, 3} 0 {2, -4} : 0 <----- 0 : {2, -3} 0 1 {2, -3} : 0 <----- QQ : {2, -2} 0 1 3 {2, -2} : QQ <--------------- QQ : {2, -1} | -1 -1 1 | 3 1 {2, -1} : QQ <---------- QQ : {2, 0} | 1 | | -1 | | 0 | {1, -3} : 0 <----- 0 : {1, -2} 0 1 {1, -2} : 0 <----- QQ : {1, -1} 0 1 1 {1, -1} : QQ <---------- QQ : {1, 0} | -1 | 1 {1, 0} : QQ <----- 0 : {1, 1} 0 {0, -2} : 0 <----- 0 : {0, -1} 0 2 {0, -1} : 0 <----- QQ : {0, 0} 0 2 {0, 0} : QQ <----- 0 : {0, 1} 0 {0, 1} : 0 <----- 0 : {0, 2} 0 {-1, -1} : 0 <----- 0 : {-1, 0} 0 o25 : SpectralSequencePageMap i26 : D^1 +---------------------+------------------------------------------+---------------------------------------+ o26 = |image | 1 0 0 0 0 0 ||image 0 |image 0 | | | 0 0 0 0 0 0 || | | | | 0 0 0 0 0 0 ||{1, 0} |{2, 0} | | | 0 1 0 0 0 0 || | | | | | | |{0, 0} | | | +---------------------+------------------------------------------+---------------------------------------+ |image 0 |subquotient (| 1 0 0 1 1 0 |, | 1 1 0 |)|subquotient (| 1 0 0 -1 1 |, | -1 1 |)| | | | 0 1 0 -1 0 0 | | -1 0 0 | | | 0 -1 1 1 0 | | 1 0 | | |{0, -1} | | 0 0 0 0 0 0 | | 0 0 0 | | | 0 1 0 -1 0 | | -1 0 | | | | | 0 0 1 0 0 1 | | 0 0 1 | | | 0 0 1 0 0 | | 0 0 | | | | | | | |{1, -1} |{2, -1} | +---------------------+------------------------------------------+---------------------------------------+ o26 : SpectralSequencePage i27 : G1 = minimalPresentation(D^1) +-------+-------+-------+ | 2 | | | o27 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o27 : SpectralSequencePage i28 : G1.dd o28 = {-2, 1} : 0 <----- 0 : {-1, 1} 0 {-2, 2} : 0 <----- 0 : {-1, 2} 0 {-2, 3} : 0 <----- 0 : {-1, 3} 0 {1, -3} : 0 <----- 0 : {2, -3} 0 {1, -2} : 0 <----- 0 : {2, -2} 0 1 {1, -1} : 0 <----- QQ : {2, -1} 0 {1, 0} : 0 <----- 0 : {2, 0} 0 {0, -2} : 0 <----- 0 : {1, -2} 0 {0, -1} : 0 <----- 0 : {1, -1} 0 2 {0, 0} : QQ <----- 0 : {1, 0} 0 {0, 1} : 0 <----- 0 : {1, 1} 0 {-1, -1} : 0 <----- 0 : {0, -1} 0 2 {-1, 0} : 0 <----- QQ : {0, 0} 0 {-1, 1} : 0 <----- 0 : {0, 1} 0 {-1, 2} : 0 <----- 0 : {0, 2} 0 {-2, 0} : 0 <----- 0 : {-1, 0} 0 o28 : SpectralSequencePageMap i29 : D^2 +------------------------------------------------+------------------------------------------------------------+-------------------------------------+ o29 = |subquotient (| 1 0 1 0 0 0 0 |, | 1 0 0 0 0 |)|subquotient (| -1 |, | -1 |) |image 0 | | | 0 0 -1 0 0 0 0 | | -1 0 0 0 0 | | | 1 | | 1 | | | | | 0 0 0 0 0 0 0 | | 0 0 0 0 0 | | | -1 | | -1 | |{2, 0} | | | 0 1 0 0 0 0 0 | | 0 0 0 0 0 | | | 0 | | 0 | | | | | | | |{0, 0} |{1, 0} | | +------------------------------------------------+------------------------------------------------------------+-------------------------------------+ |image 0 |subquotient (| 1 0 0 1 1 0 0 1 0 |, | 1 1 0 0 1 0 |)|subquotient (| 1 0 -1 1 |, | -1 1 |)| | | | 0 1 0 -1 0 1 0 0 0 | | -1 0 1 0 0 0 | | | -1 1 1 0 | | 1 0 | | |{0, -1} | | 0 0 0 0 -1 -1 1 0 0 | | 0 -1 -1 1 0 0 | | | 1 0 -1 0 | | -1 0 | | | | | 0 0 1 0 0 0 -1 0 1 | | 0 0 0 -1 0 1 | | | 0 1 0 0 | | 0 0 | | | | | | | |{1, -1} |{2, -1} | +------------------------------------------------+------------------------------------------------------------+-------------------------------------+ o29 : SpectralSequencePage i30 : G2 = minimalPresentation(D^2) +-------+-------+-------+ | 2 | | | o30 = |QQ |0 |0 | | | | | |{0, 0} |{1, 0} |{2, 0} | +-------+-------+-------+ | | | 1 | |0 |0 |QQ | | | | | |{0, -1}|{1, -1}|{2, -1}| +-------+-------+-------+ o30 : SpectralSequencePage i31 : G2.dd o31 = {-3, 2} : 0 <----- 0 : {-1, 1} 0 {-3, 3} : 0 <----- 0 : {-1, 2} 0 {-3, 4} : 0 <----- 0 : {-1, 3} 0 {0, -2} : 0 <----- 0 : {2, -3} 0 {0, -1} : 0 <----- 0 : {2, -2} 0 2 1 {0, 0} : QQ <---------- QQ : {2, -1} | 1 | | -1 | {0, 1} : 0 <----- 0 : {2, 0} 0 {-1, -1} : 0 <----- 0 : {1, -2} 0 {-1, 0} : 0 <----- 0 : {1, -1} 0 {-1, 1} : 0 <----- 0 : {1, 0} 0 {-1, 2} : 0 <----- 0 : {1, 1} 0 {-2, 0} : 0 <----- 0 : {0, -1} 0 2 {-2, 1} : 0 <----- QQ : {0, 0} 0 {-2, 2} : 0 <----- 0 : {0, 1} 0 {-2, 3} : 0 <----- 0 : {0, 2} 0 {-3, 1} : 0 <----- 0 : {-1, 0} 0 o31 : SpectralSequencePageMap i32 : D^infinity +------------------------------------------------------------------+ o32 = |subquotient (| 1 0 1 1 0 0 0 0 0 0 |, | 1 1 0 0 0 0 0 0 |)| | | 0 0 -1 0 1 0 0 0 0 0 | | -1 0 1 0 0 0 0 0 | | | | 0 0 0 -1 -1 1 0 0 0 0 | | 0 -1 -1 1 0 0 0 0 | | | | 0 1 0 0 0 -1 0 0 0 0 | | 0 0 0 -1 0 0 0 0 | | | | |{0, 0} | +------------------------------------------------------------------+ o32 : Page