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
|