group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
What is called effective homology is the study of (effective) algorithms for computing invariants in algebraic topology, such as homology groups but also homotopy groups.
From the first part of “Effective homology, a survey”, we can see why effective homology is useful.
It’s much better to consider only the simply-connected spaces. We have two methods
Sullivan’s method is more restrictive (to Q) but much more efficient than Brown’s method. Brown’s method is not practical at all, even with the most powerful computer at hand. Nevertheless, even Sullivan’s method is at least #P-hard.
That’s how effective method kicks in. It adapts Hirsch’s method [fn:3]. Using functional programming, this becomes a real computing tool for homology and homotopy groups.
Francis Sergeraert, Effective homology, a survey (pdf)
[fn:1][20] Dennis Sullivan. Infinitesimal calculations in topology.
[fn:2][3] Edgar H. Brown Jr.. Finite computability of Postnikov complexes. Annals of Mathematics, 1957, vol. 65, pp 1-20.
[fn:3][11] G. Hirsch. Sur les groupes d’homologie des espaces fibr´es.