Free loop spaces and computability in algebraic topology. (English) Zbl 0687.55013
Differential geometry, Proc. 6th Int. Colloq., Santiago de Compostela/Spain 1988, Cursos Congr. Univ. Santiago de Compostela 61, 239-245 (1989).
[For the entire collection see Zbl 0682.00012.]
This paper describes without proof how to construct Turing algorithms for such problems as: given a simply connected finite simplicial complex X, compute the homology of its free loop space \(\Omega\) X. There are two main steps. The first passes algorithmically from a “functional code” (a finite description) for X to a functional code for \(\Omega\) X. The second shows that the spectral sequence machinery for homology in fibrations can be converted into algorithms, provided that one replaces the usual homology groups by “effective homology”, this being a functor which has a functional code and from which the usual homology groups can be computed.
This paper describes without proof how to construct Turing algorithms for such problems as: given a simply connected finite simplicial complex X, compute the homology of its free loop space \(\Omega\) X. There are two main steps. The first passes algorithmically from a “functional code” (a finite description) for X to a functional code for \(\Omega\) X. The second shows that the spectral sequence machinery for homology in fibrations can be converted into algorithms, provided that one replaces the usual homology groups by “effective homology”, this being a functor which has a functional code and from which the usual homology groups can be computed.
Reviewer: R.J.Steiner
MSC:
| 55T10 | Serre spectral sequences |
| 55T20 | Eilenberg-Moore spectral sequences |
| 03D60 | Computability and recursion theory on ordinals, admissible sets, etc. |
