Homologie effective. II. (Effective homology. II). (French) Zbl 0614.55011
In this second part the author describes the algorithms provided by the effective homology introduced in the first part of his article [ibid. 304, 279-282 (1987; Zbl 0608.55004)] when they are applied to fibrations. The author considers the case of simplicial fibrations in the sense of J. P. May [Simplicial objects in algebraic topology (1968; Zbl 0165.260)]. Then the main results are the following theorems:
Theorem 1. Three algorithms can be constructed: \(Entering:=(\Phi,{\mathcal H}B_{\Phi},{\mathcal H}F_{\Phi})\), resp. (\(\Phi\),\({\mathcal H}B_{\Phi},{\mathcal H}E_{\Phi})\), (\(\Phi\),\({\mathcal H}E_{\Phi},{\mathcal H}F_{\Phi})\); \(Emergence:={\mathcal H}E_{\Phi}\) (resp. \({\mathcal H}F_{\Phi}\), \({\mathcal H}B_{\Phi})\); where (a) \(\Phi\) is a simplicial fibration; (b) \({\mathcal H}B_{\Phi}\), \({\mathcal H}E_{\Phi}\) and \({\mathcal H}F_{\Phi}\) are the effective homologies of \(B_{\Phi}\), \(E_{\Phi}\) and \(F_{\Phi}.\)
Theorem 2. An algorithm can be constructed: \(Entering:=(\Phi,{\mathcal H}B_{\Phi},{\mathcal H}F_{\Phi})\); \(Emergence:=(d_{\tau},h)\); where (a) the notations for the entering are the same as in Theorem 1; (b) \(d_{\tau}\) is a differential on the tensor product \(HB_*\otimes HF_*\) where \(HB_*\) is the ”image” of \({\mathcal H}B_{\Phi}\) and the same for \(HF_*\); (c) h is a homotopy equivalence between \(E_*\), the chain complex associated to \(E_{\Phi}\), and \((HB_*\otimes HF_*,d_{\tau}).\)
Theorem 3. An algorithm can be constructed: \(Entering:=(G,{\mathcal H}G)\); \(Emergence:={\mathcal H}BG\); where (a) G is a simplicial connected group; (b) \({\mathcal H}G\) is the effective homology supposed as being defined; (c) BG is the classifying space of G.
Theorem 6. An algorithm can be constructed: \(Entering:=(C,{\mathcal H}C)\); \(Emergence:={\mathcal H}\Omega C\); where (a) C is a simple connected simplicial set; (b) \({\mathcal H}C\) is its effective homology supposed as being defined; (c) \(\Omega\) C is the simplicial version of the loop space of C (the construction G of D. M. Kan [Ann. Math., II. Ser. 67, 282-312 (1958; Zbl 0091.369)]).
These results and three other corollaries represent the ”effective homology” versions of the spectral sequence of Serre and the Postnikov tower of E on the author’s Turing machine. The author specifies that analogous results can be obtained for the Eilenberg-MacLane homology of discrete groups, for the Lie Z-algebras, etc.
Theorem 1. Three algorithms can be constructed: \(Entering:=(\Phi,{\mathcal H}B_{\Phi},{\mathcal H}F_{\Phi})\), resp. (\(\Phi\),\({\mathcal H}B_{\Phi},{\mathcal H}E_{\Phi})\), (\(\Phi\),\({\mathcal H}E_{\Phi},{\mathcal H}F_{\Phi})\); \(Emergence:={\mathcal H}E_{\Phi}\) (resp. \({\mathcal H}F_{\Phi}\), \({\mathcal H}B_{\Phi})\); where (a) \(\Phi\) is a simplicial fibration; (b) \({\mathcal H}B_{\Phi}\), \({\mathcal H}E_{\Phi}\) and \({\mathcal H}F_{\Phi}\) are the effective homologies of \(B_{\Phi}\), \(E_{\Phi}\) and \(F_{\Phi}.\)
Theorem 2. An algorithm can be constructed: \(Entering:=(\Phi,{\mathcal H}B_{\Phi},{\mathcal H}F_{\Phi})\); \(Emergence:=(d_{\tau},h)\); where (a) the notations for the entering are the same as in Theorem 1; (b) \(d_{\tau}\) is a differential on the tensor product \(HB_*\otimes HF_*\) where \(HB_*\) is the ”image” of \({\mathcal H}B_{\Phi}\) and the same for \(HF_*\); (c) h is a homotopy equivalence between \(E_*\), the chain complex associated to \(E_{\Phi}\), and \((HB_*\otimes HF_*,d_{\tau}).\)
Theorem 3. An algorithm can be constructed: \(Entering:=(G,{\mathcal H}G)\); \(Emergence:={\mathcal H}BG\); where (a) G is a simplicial connected group; (b) \({\mathcal H}G\) is the effective homology supposed as being defined; (c) BG is the classifying space of G.
Theorem 6. An algorithm can be constructed: \(Entering:=(C,{\mathcal H}C)\); \(Emergence:={\mathcal H}\Omega C\); where (a) C is a simple connected simplicial set; (b) \({\mathcal H}C\) is its effective homology supposed as being defined; (c) \(\Omega\) C is the simplicial version of the loop space of C (the construction G of D. M. Kan [Ann. Math., II. Ser. 67, 282-312 (1958; Zbl 0091.369)]).
These results and three other corollaries represent the ”effective homology” versions of the spectral sequence of Serre and the Postnikov tower of E on the author’s Turing machine. The author specifies that analogous results can be obtained for the Eilenberg-MacLane homology of discrete groups, for the Lie Z-algebras, etc.
Reviewer: Ioan Pop (Iaşi)
MSC:
| 55R20 | Spectral sequences and homology of fiber spaces in algebraic topology |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
| 57T25 | Homology and cohomology of \(H\)-spaces |
| 55P35 | Loop spaces |
| 68Q05 | Models of computation (Turing machines, etc.) (MSC2010) |
