In this example we compute the spectral sequence arising from the quotient map $\mathbb{S}^2 \rightarrow \mathbb{R} \mathbb{P}^2$, given by identifying anti-podal points. This map can be realized by a simplicial map along the lines of Exercise 27, Section 6.5 of Armstrong's book Basic Topology. In order to give a combinatorial picture of the quotient map $\mathbb{S}^2 \rightarrow \mathbb{R} \mathbb{P}^2$, given by identifying anti-podal points, we first make an appropriate simplicial realization of $\mathbb{S}^2$. Note that we have added a few barycentric coordinates.
i1 : S = ZZ[v1,v2,v3,v4,v5,v6,v15,v12,v36,v34,v46,v25]; |
i2 : twoSphere = simplicialComplex {v3*v4*v5, v5*v4*v15, v15*v34*v4, v15*v34*v1, v34*v1*v6, v34*v46*v6, v36*v46*v6, v3*v4*v46, v4*v46*v34, v3*v46*v36, v1*v6*v2, v6*v2*v36, v2*v36*v12,v36*v12*v3, v12*v3*v5, v12*v5*v25, v25*v5*v15, v2*v12*v25, v1*v2*v25, v1*v25*v15};
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We can check that the homology of the simplicial complex twoSphere agrees with that of $\mathbb{S}^2$.
i3 : C = truncate(chainComplex twoSphere,1)
12 30 20
o3 = image 0 <-- ZZ <-- ZZ <-- ZZ
-1 0 1 2
o3 : ChainComplex
|
i4 : prune HH C
o4 = -1 : 0
1
0 : ZZ
1 : 0
1
2 : ZZ
o4 : GradedModule
|
We now write down our simplicial complex whose topological realization is $\mathbb{R} \mathbb{P}^2$.
i5 : R = ZZ[a,b,c,d,e,f]; |
i6 : realProjectivePlane = simplicialComplex {a*b*c, b*c*d, c*d*e, a*e*d, e*b*a, e*f*b, d*f*b, a*f*d, c*f*e,a*f*c};
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Again we can check that we've entered a simplicial complex whose homology agrees with that of the real projective plane.
i7 : B = truncate(chainComplex realProjectivePlane,1)
6 15 10
o7 = image 0 <-- ZZ <-- ZZ <-- ZZ
-1 0 1 2
o7 : ChainComplex
|
i8 : prune HH B
o8 = -1 : 0
1
0 : ZZ
1 : cokernel | 2 |
2 : 0
o8 : GradedModule
|
We now compute the fibers of the anti-podal quoitent map $\mathbb{S}^2 \rightarrow \mathbb{R} \mathbb{P}^2$. The way this works for example is: $a = v3 ~ v1, b = v6 ~ v5, d = v36 ~ v15, c = v4 ~ v2, e = v34 ~ v12, f = v46 ~ v25$
The fibers over the vertices of $\mathbb{R} \mathbb{P}^2$ are:
i9 : F0twoSphere = simplicialComplex {v1,v3,v5,v6, v4,v2, v36,v15, v34,v12, v46,v25}
o9 = | v25 v46 v34 v36 v12 v15 v6 v5 v4 v3 v2 v1 |
o9 : SimplicialComplex
|
The fibers over the edges of $\mathbb{R}\mathbb{P}^2$ are:
i10 : F1twoSphere = simplicialComplex {v3*v4, v1*v2,v3*v5, v1*v6,v4*v5, v2*v6, v5*v15, v6*v36, v4*v34, v2*v12, v15*v34, v36*v12, v1*v15, v3*v36, v46*v34, v25*v12, v6*v34, v5*v12, v6*v46, v5*v25, v36*v46, v15*v25, v3*v46, v1*v25, v4*v15, v2*v36, v1*v34, v3*v12, v4*v46, v25*v2}
o10 = | v12v25 v15v25 v5v25 v2v25 v1v25 v34v46 v36v46 v6v46 v4v46 v3v46 v15v34 v6v34 v4v34 v1v34 v12v36 v6v36 v3v36 v2v36 v5v12 v3v12 v2v12 v5v15 v4v15 v1v15 v2v6 v1v6 v4v5 v3v5 v3v4 v1v2 |
o10 : SimplicialComplex
|
The fibers over the faces is all of $\mathbb{S}^2$.
i11 : F2twoSphere = twoSphere o11 = | v5v12v25 v2v12v25 v5v15v25 v1v15v25 v1v2v25 v6v34v46 v4v34v46 v6v36v46 v3v36v46 v3v4v46 v4v15v34 v1v15v34 v1v6v34 v3v12v36 v2v12v36 v2v6v36 v3v5v12 v4v5v15 v1v2v6 v3v4v5 | o11 : SimplicialComplex |
The resulting filtered complex is:
i12 : K = filteredComplex({F2twoSphere, F1twoSphere, F0twoSphere}, ReducedHomology => false)
o12 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0
-1 0 1 2
0 : image 0 <-- image | 1 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0
| 0 1 0 0 0 0 0 0 0 0 0 0 |
-1 | 0 0 1 0 0 0 0 0 0 0 0 0 | 1 2
| 0 0 0 1 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 1 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 1 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 1 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 1 |
0
1 : image 0 <-- image | 1 0 0 0 0 0 0 0 0 0 0 0 | <-- image | 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 | <-- image 0
| 0 1 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 |
-1 | 0 0 1 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 | 2
| 0 0 0 1 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 1 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 1 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 1 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 1 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 1 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 1 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 1 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 1 | | 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 1 |
1
12 30 20
2 : image 0 <-- ZZ <-- ZZ <-- ZZ
-1 0 1 2
o12 : FilteredComplex
|
We now compute the resulting spectral sequence.
i13 : E = prune spectralSequence K o13 = E o13 : SpectralSequence |
i14 : E^0
+------+------+------+
| 12 | 30 | 20 |
o14 = |ZZ |ZZ |ZZ |
| | | |
|{0, 0}|{1, 0}|{2, 0}|
+------+------+------+
o14 : SpectralSequencePage
|
i15 : E^1
+------+------+------+
| 12 | 30 | 20 |
o15 = |ZZ |ZZ |ZZ |
| | | |
|{0, 0}|{1, 0}|{2, 0}|
+------+------+------+
o15 : SpectralSequencePage
|
i16 : E^0 .dd
o16 = {-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
{2, -3} : 0 <----- 0 : {2, -2}
0
{2, -2} : 0 <----- 0 : {2, -1}
0
20
{2, -1} : 0 <----- ZZ : {2, 0}
0
{1, -3} : 0 <----- 0 : {1, -2}
0
{1, -2} : 0 <----- 0 : {1, -1}
0
30
{1, -1} : 0 <----- ZZ : {1, 0}
0
30
{1, 0} : ZZ <----- 0 : {1, 1}
0
{0, -2} : 0 <----- 0 : {0, -1}
0
12
{0, -1} : 0 <----- ZZ : {0, 0}
0
12
{0, 0} : ZZ <----- 0 : {0, 1}
0
{0, 1} : 0 <----- 0 : {0, 2}
0
{-1, -1} : 0 <----- 0 : {-1, 0}
0
o16 : SpectralSequencePageMap
|
i17 : E^1 .dd
o17 = {-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} : 0 <----- 0 : {2, -1}
0
30 20
{1, 0} : ZZ <------------------------------------------------------------------- ZZ : {2, 0}
| -1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 1 0 -1 0 0 0 0 0 0 0 0 0 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 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 1 0 0 1 0 0 0 0 0 0 0 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 0 0 0 0 0 -1 -1 0 0 0 0 0 0 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 -1 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 -1 -1 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 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 1 -1 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 |
| 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 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 1 -1 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 -1 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 1 1 0 0 |
| 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 |
| 0 0 -1 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 1 1 |
| 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 |
| 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 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 -1 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 |
{0, -2} : 0 <----- 0 : {1, -2}
0
{0, -1} : 0 <----- 0 : {1, -1}
0
12 30
{0, 0} : ZZ <------------------------------------------------------------------------------------------------- ZZ : {1, 0}
| 1 1 1 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 |
| -1 0 0 0 0 1 1 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 1 1 1 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 -1 0 0 0 0 1 1 1 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 -1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 |
| 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 |
| 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 -1 0 0 0 0 0 1 1 0 0 0 0 |
| 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 1 0 0 |
| 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 1 0 |
| 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 -1 0 0 0 0 1 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 -1 -1 |
| 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 -1 0 0 |
{0, 1} : 0 <----- 0 : {1, 1}
0
{-1, -1} : 0 <----- 0 : {0, -1}
0
12
{-1, 0} : 0 <----- ZZ : {0, 0}
0
{-1, 1} : 0 <----- 0 : {0, 1}
0
{-1, 2} : 0 <----- 0 : {0, 2}
0
{-2, 0} : 0 <----- 0 : {-1, 0}
0
o17 : SpectralSequencePageMap
|
i18 : E^2
+------+------+------+
| 1 | | 1 |
o18 = |ZZ |0 |ZZ |
| | | |
|{0, 0}|{1, 0}|{2, 0}|
+------+------+------+
o18 : SpectralSequencePage
|
i19 : E^2 .dd
o19 = {-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
{0, 0} : ZZ <----- 0 : {2, -1}
0
1
{0, 1} : 0 <----- ZZ : {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 <----- ZZ : {0, 0}
0
{-2, 2} : 0 <----- 0 : {0, 1}
0
{-2, 3} : 0 <----- 0 : {0, 2}
0
{-3, 1} : 0 <----- 0 : {-1, 0}
0
o19 : SpectralSequencePageMap
|