Relative homology

[mglisse]

Since several users have asked: we should provide an example showing how to use persistence to compute the relative homology of a complex and a subcomplex (mostly how to interpret the result).

[MathieuCarriere]

I agree. Extended persistence could also be interesting since I have received a few messages from people trying to use Gudhi for computing extended persistence.

[VincentRouvreau]

Was this issue fixed by #215 ? @mglisse @MathieuCarriere

[mglisse]

I don't think #215 contains an example showing how to compute the relative homology of a pair (L,K), so while it is related, it answers a different question.

[mglisse]

In a discussion, we mentioned the possibility of adding an API relative_betti_numbers(x,y) to compute the relative homology of (K_y, K_x).

[shilangyu]

Has there been any progress in this space? If not, what are the possible approaches to compute relative homology of a complex and its subcomplex using the current API? I'd be happy to contribute documentation upon some guidance.

[mglisse]

The simplest approach, when you have a small complex S included in a big complex B and you want their relative homology, is to construct the big complex, assign a filtration value of 0 to the cells of S and 1 to the others, and compute persistent homology. The relative homology is then just a matter of interpreting the diagram:

• intervals [0, inf) don't matter
• intervals [1, inf) count normally
• intervals [0,1) have a dimension shift (an edge that kills a connected component in persistent homology actually creates a cycle in relative homology)

Or something similar, I don't promise that I didn't write this too quickly.