Standard homological algebra is not constructive, and this is frequently the source of serious problems when algorithms are looked for. In particular the usual exact and spectral sequences of homological algebra frequently are not sufficient to obtain some unknown homology or homotopy group. We will explain it is not difficult to fill in this gap, the main tools being on one hand, from a mathematical point of view, the so-called Homological Perturbation Lemma, and on the other hand, from a computational point of view, Functional Programming.
We will illustrate this area of constructive mathematics by applications in two domains:
a) Commutative Algebra frequently meets homological objects, in particular when resolutions are involved (syzygies).
Constructive Homological Algebra produces new methods to process old problems such as homology of Koszul complexes, resolutions, minimal resolutions. The solutions so obtained are constructive and therefore more complete thant the usual ones, an important point for their concrete use.
b) Algebraic Topology is the historical origin of Homological Algebra. The usual exact and spectral sequences of Algebraic Topology can be easily transformed into new effective versions, giving algorithms computing for example unknown homology and homotopy groups. In particular the effective version of the Eilenberg-Moore spectral sequence gives a very simple solution for the old Adams' problem: what algorithm could compute the homology groups of iterated loop spaces?
Machine programs and machine demonstrations will also concretely illustrate the theoretical results so obtained.
Plan:
1. Elements of Homological Algebra. Ordinary Homological Algebra is not constructive.
2. Notion of (constructive) "Solution of the Homological Problem for a chain-complex". Reductions between chain-complexes. Homological Perturbation Lemma. Particular case of twisted sums (cone construction).
3. Applications in Commutative Algebra: Effective Homology of Koszul complexes. Equivalence between *effective* homology of Koszul complexes and resolutions. A simple solution for a minimal resolution.
4. Twisted products in Algebraic Topology. The effective version of the Serre spectral sequence. Application: a theoretical and concrete algorithm computing homotopy groups.
5. Twisted divisions in Algebraic Topology. The effective version of the Eilenberg-Moore spectral sequence. Application: a theoretical and concrete algorithm solving Adams' problem for iterated loop spaces.
6. Effective Algebraic Topology and Postnikov invariants. Conclusion: the pretended Postnikov invariants in fact are not... invariant!