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Question. Let $(C_\bullet,\partial)$ be a $\mathbb{Z}/2\mathbb{Z}$-chain complex generated by $a$, $b_1,b_2,b_3$, $c_1,c_2$ with gradings: $$|a|=2,|b_1|=|b_2|=|b_3|=1,|c_1|=|c_2|=0.$$ Which graded modules (polynomial) can be realized as (the Poincaré polynomial of) $H_\bullet(C_\bullet,\partial)$?

I can always do a case by case computation, but I am exhausted just thinking about it, as we know that: $$\begin{align}\partial a&\in\{0,b_1,b_2,b_3,b_1+b_2,b_1+b_3,b_2+b_3,b_1+b_2+b_3\},\\\partial b_1,\partial b_2,\partial b_3&\in\{0,c_1,c_2,c_1+c_2\},\\\partial c_1,\partial c_2&=0,\end{align}$$ there is $512$ different $\partial_i$, but some of them will not satisfy $\partial_i\circ\partial_i=0$.

  • Is there a clever way to carry on the computation?
  • Can a computer algebra software, as SageMath, solve my problem?

Any enlightenment will be greatly appreciated.

Context. I have a chain complex $(C_\bullet,\partial)$ coming from a geometric setting, for which I know $C_\bullet$ as a graded module (I am able to compute its generators and their grading), but whose boundary map $\partial$ is unknown to me (and I am actually interested in a lot of different boundary map on $C$).

I want to find all chain complex $(C_\bullet,\partial_i)$ and compute their homology, said explicitely:

  • the graded module $C_\bullet$ is fixed,
  • I look for endomorphisms $\partial_i$ with $\partial_i\colon C_\bullet\to C_{\bullet-1}$ and $\partial_i\circ\partial_i=0$,
  • I want to compute $H_\bullet(C_\bullet,\partial_i)=\ker(\partial_i)/\textrm{im}(\partial_i)$ as a graded module.

With sufficient partial information on the boundary map $\partial$, I hope to be able to compute the homology of $(C_\bullet,\partial)$ guessing which of the $H_\bullet(C_\bullet,\partial_i)$ it is.

Precise context. I want to compute all the mixed generating family homologies of the Hopf link with Maslov potential from the lower strand to the upper strand being $0,1,1,2$.

In general, it is really hard to compute the boundary map of the mixed generating family homology, as it basically involves knowing the gradient flow lines of a qualitatively constructed map and structural results on this homology are still to be discovered.

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You can do it in SageMath as follows, if I understood it correctly:

var('t')
poincare_polynomials = set([])
import itertools
for (a_to_b, b_to_c) in itertools.product(itertools.product(GF(2),repeat=3), itertools.product(GF(2),repeat=6)):
    try:
        C = ChainComplex({2: matrix(GF(2), 3, 1, a_to_b), 1: matrix(GF(2), 2, 3, b_to_c)}, degree=-1)
        #print C.homology(generators=True)
        #P_C = sum(C.free_module_rank(deg)*t^deg for deg in C.nonzero_degrees())
        P_H = sum(C.homology(deg).dimension()*t^deg for deg in C.nonzero_degrees())
        poincare_polynomials.add(P_H)
        #print ((P_C - P_H)/(1+t)).full_simplify()
    except ValueError:
        # differentials not compatible
        pass
print poincare_polynomials

which outputs

set([0, 2*t + 2, t^2 + t, t^2 + 3*t + 2, t^2 + 2*t + 1, t + 1])

You can modify the code to get any additional information you want. See Chain complexes in the Sage reference manual.

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  • 1
    $\begingroup$ Your code seems to do the job, the outputs match my expectation! It helps a lot, thank you! $\endgroup$ – C. Falcon Jan 26 at 23:08

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