nLab
effective homology

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Constructivism, Realizability, Computability

Contents

Idea

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

  1. Sullivan method for rational homotopy (based on minimal model theory) [fn:1]
  2. Edgar Brown method for integer homotopy. [fn:2]

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.

References