homologyIsomorphism(SpectralSequence,ZZ,ZZ,ZZ) -- the homology isomorphism

Synopsis

Function: homologyIsomorphism
Usage:

g = homologyIsomorphism(SpectralSequence, ZZ, ZZ, ZZ)
Inputs:
- E, a spectral sequence,
- p, an integer,
- q, an integer,
- r, an integer,
Outputs:
- g, a matrix,

Description

Computes the isomorphism $ker d^r_{p,q} / image d^r_{p + r, q - r + 1} \rightarrow E^{r+1}_{p,q}$

i1 : A = ZZ [s,t,u,v,w] ;

i2 : K = filteredComplex(reverse {simplicialComplex {s}, simplicialComplex {s,t}, simplicialComplex {s,t,u}, simplicialComplex {s*t, u}, simplicialComplex {s*t, u, v}, simplicialComplex {s*t, u, v, w}, simplicialComplex {s*t, s*w ,u, v}, simplicialComplex {s*t, s*w ,t * w, u, v}, simplicialComplex {s*t, s*w ,t * w, u * v}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u, u * w}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u, u * w, t* u}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u, u * w, t* u, t*u*w}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u, u * w, t* u, t*u*w, s*u*w}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u, u * w, t* u, t*u*w, s*u*w,s*t*u}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u, u * w, t* u, t*u*w, s*u*w,s*t*u, s*u*v}, simplicialComplex {s*t, s*w ,t * w, u * v, s * v, s*u, u * w, t* u, t*u*w, s*u*w,s*t*u, s*u*v, s*t*w}}, ReducedHomology => false);

i3 : E = prune spectralSequence K

o3 = E

o3 : SpectralSequence

i4 : e = spectralSequence K

o4 = e

o4 : SpectralSequence

i5 : apply(keys support E^11, i -> homologyIsomorphism(E, i#0, i#1, 11))

o5 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, | 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}

o5 : List

i6 : apply(keys support e^11, i -> homologyIsomorphism(e, i#0, i#1, 11))

o6 = {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 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, | 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 0 0 0 0 0 0 0 0 0 0 0 |
     0 |           | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
                   | 0 0 0 0 0 0 0 0 0 0 0 0 |
     ------------------------------------------------------------------------
     0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}

o6 : List

Ways to use this method: