Effective algebraic topology.

*(English)*Zbl 0731.55015
Mem. Am. Math. Soc. 451, 63 p. (1991).

According to the author, this work gives the first four of a series of papers which provide means for effective solutions to computation, construction, and decision problems in algebraic topology.

The first paper is entitled “A five lemma for calculations in homological algebra”. Generally, exact sequences in homological algebra do not give complete information about an object one wants to calculate, but this note shows that at least in principle full information can be received, if homological algebra is restricted to finitely generated abelian groups and the notion of calculability is substantially strengthened. Roughly speaking, the main result is that the middle term of a 5-term exact sequence is calculable, if the other four are. The second paper, “Fibrations with calculable homology”, has as a main consequence the following fundamental calculation theorem for fiber- spaces: Each homology group of one of the complexes F, X, Y of a fibration \(F\to X\to Y\) is extended calculable, if each homology group of the two other complexes is extended calculable. Applications are given to loop spaces and rational homotopy groups. The third paper, “An algorithm for calculating homotopy groups” shows the effective calculability of each homotopy group \(\pi_ i(X)\), \(i\geq 1\), X a simply connected complex of finite type, via the homology of iterated loop spaces. The fourth note, “The effective computability of k-invariants” proves the effective computability of each k-invariant \(k\in F(n,X)\), \(n\geq 1\), of a simply connected, finitely generated complex X; n.X its n-th Postnikov stage. \(F(U)=H^{n+1}(U;TG)\), TG the type of the homotopy group \(\pi_{n+1}(X)\). More precisely, an algorithm is provided which constructs two finitely generated subcomplexes \(n_ 1\subset n_ 2\subset n.X\) and a cohomology class ḵ\(\in F(n_ 1)\) such that the restriction \(F(n,X)\to F(n_ 1)\) induces an isomorphism \(F(n,X)\to^{\simeq}image(F(n_ 2)\to F(n_ 1))\) which maps the classical k-invariant \(k\in F(n,X)\) onto ḵ\(\in F(n_ 1).\)

Summarizing, we can say that the novelty of this work consists in a new handling of exact sequences in homological algebra and in algebraic topology.

The first paper is entitled “A five lemma for calculations in homological algebra”. Generally, exact sequences in homological algebra do not give complete information about an object one wants to calculate, but this note shows that at least in principle full information can be received, if homological algebra is restricted to finitely generated abelian groups and the notion of calculability is substantially strengthened. Roughly speaking, the main result is that the middle term of a 5-term exact sequence is calculable, if the other four are. The second paper, “Fibrations with calculable homology”, has as a main consequence the following fundamental calculation theorem for fiber- spaces: Each homology group of one of the complexes F, X, Y of a fibration \(F\to X\to Y\) is extended calculable, if each homology group of the two other complexes is extended calculable. Applications are given to loop spaces and rational homotopy groups. The third paper, “An algorithm for calculating homotopy groups” shows the effective calculability of each homotopy group \(\pi_ i(X)\), \(i\geq 1\), X a simply connected complex of finite type, via the homology of iterated loop spaces. The fourth note, “The effective computability of k-invariants” proves the effective computability of each k-invariant \(k\in F(n,X)\), \(n\geq 1\), of a simply connected, finitely generated complex X; n.X its n-th Postnikov stage. \(F(U)=H^{n+1}(U;TG)\), TG the type of the homotopy group \(\pi_{n+1}(X)\). More precisely, an algorithm is provided which constructs two finitely generated subcomplexes \(n_ 1\subset n_ 2\subset n.X\) and a cohomology class ḵ\(\in F(n_ 1)\) such that the restriction \(F(n,X)\to F(n_ 1)\) induces an isomorphism \(F(n,X)\to^{\simeq}image(F(n_ 2)\to F(n_ 1))\) which maps the classical k-invariant \(k\in F(n,X)\) onto ḵ\(\in F(n_ 1).\)

Summarizing, we can say that the novelty of this work consists in a new handling of exact sequences in homological algebra and in algebraic topology.

Reviewer: I.Pop (Iaşi)

##### MSC:

55T10 | Serre spectral sequences |

55S45 | Postnikov systems, \(k\)-invariants |

55Q05 | Homotopy groups, general; sets of homotopy classes |