Suppose X is an abstract simplicial complex whose vertices are integers. We specify X by listing its facets:
i1 : Xfacets = {{1,2},{2,3},{3,4},{1,4},{1,3}};
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In this example, X is one-dimensional. It is a square with vertices {1,2,3,4} and with a diagonal edge connecting 1 to 3.
To specify a filtration of X, we write a function that takes a face to its filtration degree. In this example, we set each vertex to have degree 0, the square to have degree 1, and the diagonal edge {1,3} to have degree 2.
i2 : filt = f -> if #f == 1 then 0 else if f == {1,3} then 2 else 1;
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We can now build the exact couple and view the corresponding spectral sequence.
i3 : couple = filteredSimplicialComplexCouple(Xfacets, filt);
warning: clearing value of symbol t to allow access to subscripted variables based on it
: debug with expression debug 5504 or with command line option --debug 5504
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i4 : plotPages((-1..3,-1..3,1..2), prune @@ evaluateInDegree, couple)
warning: clearing value of symbol e to allow access to subscripted variables based on it
: debug with expression debug 3903 or with command line option --debug 3903
warning: clearing value of symbol t to allow access to subscripted variables based on it
: debug with expression debug 5504 or with command line option --debug 5504
page 1, with differential of degree {-1, -1}:
+----++----+---+---+---+---+
|q=3 ||0 |0 |0 |0 |0 |
+----++----+---+---+---+---+
| || | | 1| | |
|q=2 ||0 |0 |ZZ |0 |0 |
+----++----+---+---+---+---+
| || | | 4| | |
|q=1 ||0 |0 |ZZ |0 |0 |
+----++----+---+---+---+---+
| || | 4| | | |
|q=0 ||0 |ZZ |0 |0 |0 |
+----++----+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |
+----++----+---+---+---+---+
| ||p=-1|p=0|p=1|p=2|p=3|
+----++----+---+---+---+---+
page 2, with differential of degree {-1, -2}:
+----++----+---+---+---+---+
|q=3 ||0 |0 |0 |0 |0 |
+----++----+---+---+---+---+
| || | | 1| | |
|q=2 ||0 |0 |ZZ |0 |0 |
+----++----+---+---+---+---+
| || | | 1| | |
|q=1 ||0 |0 |ZZ |0 |0 |
+----++----+---+---+---+---+
| || | 1| | | |
|q=0 ||0 |ZZ |0 |0 |0 |
+----++----+---+---+---+---+
|q=-1||0 |0 |0 |0 |0 |
+----++----+---+---+---+---+
| ||p=-1|p=0|p=1|p=2|p=3|
+----++----+---+---+---+---+
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Here is how to interpret the output. The "p" coordinate is homological degree, and the "q" coordinate is filtration degree.
On page 1, we see four 0-dimensional classes in filtration degree 0, and five 1-dimensional classes spread between filtration degree 0 and 1. The differential on page 1 has bidegree $(-1,-1)$.
On page 2, we see that three 1-dimensional classes have cancelled three 0-dimensional classes, and we are left with fewer classes. The differential now has bidegree $(-1,-2)$.
In fact, the sequence is converged: column $p$ now gives the associated graded of the filtration on $H_p$.
The filtration must be increasing: if $A, B \subseteq X$ are faces with $A \subseteq B$, then the filtration degree of $A$ must not exceed the filtration degree of $B$.
The object filteredSimplicialComplexCouple is a method function.