Obtaining symbolic generators from homology()

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From your question it is not completely clear to me what you expect, since i guess you do not want to play with elements of the symbolic ring (which is what "symbol" usually refers to in Sage), so i suspect you want to give Python names to the elements of your graded basis. Here is a suggestion, please explicit more if it is not satisfactory.

When you write:

sage: H = TorusComplex.homology(generators=true) ; H
{0: [(Vector space of dimension 1 over Ring of integers modulo 2,
   Chain(0:(1)))],
 1: [(Vector space of dimension 1 over Ring of integers modulo 2,
   Chain(1:(1, 0))),
  (Vector space of dimension 1 over Ring of integers modulo 2,
   Chain(1:(0, 1)))],
 2: [(Vector space of dimension 1 over Ring of integers modulo 2,
   Chain(2:(1)))]}

H is a dictionary that maps a dimension (an element of the set {0,1,2}) to a list of 2-uples (vector space, generator). So, you need to recover the list of generators that appear as second projection of the values of this dictionary. Here is a possible way:

sage: D = dict()

sage: for dim in H:
....:     for n in range(len(H[dim])):
....:         D[dim,n] = H[dim][n][1]

sage: D
{(0, 0): Chain(0:(1)),
 (1, 0): Chain(1:(1, 0)),
 (1, 1): Chain(1:(0, 1)),
 (2, 0): Chain(2:(1))}

Now you can easily give Python names to those four chains, that form your graded basis:

sage: a,b,c,d = D[0,0], D[1,0], D[1,1], D[2,0]

Then you can play with them:

sage: a+a
Trivial chain

sage: a+b+a == b
True

sage: a+b
Chain with 2 nonzero terms over Ring of integers modulo 2

sage: b+c
Chain(1:(1, 1))

sage: (a+b+2*c+d)._vec
{0: (1), 1: (1, 0), 2: (1)}

sage: ascii_art(a+b+2*c+d)
   d_2       d_1       d_0       d_-1  
0 <---- [1] <---- [1] <---- [1] <----- 0
                  [0]

As you can see, an element of the homology is stored as dictionary that map each dimension to a vector in the corresponding vector space, which is somehow richer than flat names such as a,b,c,d since it keeps the graded information.