This is a very good question.

An approach to using algebraic topology in conjunction with quantum entanglement is given in

D. Melnikova,, A. Mironovb, S. Mironovb, A. Morozovb and An. Morozovb,

From Topological to Quantum Entanglement :

https://arxiv.org/pdf/1809.04574.pdf

To get started, see the attached image from page 3 of the paper by D. Melnikova et al. It is observed (by D. Melnikova et al.) that one can glue the topological spaces along boundaries, which differ by orientation, to form new topological spaces.