Homotopy classification of maps between \(r-1\) connected \(2r\) dimensional manifolds. (English) Zbl 1132.55002
The present work studies the classification of the homotopy class of maps between two \(r-1\) connected \(2r\) dimensional manifolds. These manifolds have the homotopy type of complexes which have a cell structure with one cell in dimension zero, a number \(n\) \((n>0)\) of cells in dimension \(r\) \((r>1)\) and one cell in dimension \(2r\). The authors give a description of the set of homotopy classes of maps between two such manifolds which relies on the knowledge of the homotopy groups of a sphere in dimensions up to the double of the dimension of the sphere. Further, an algorithm to compute those sets is provided and some explicit calculations in low dimensions are given. The authors obtain their results by means of the Puppe exact sequence. They work in the part of the sequence where the sets involved are not necessarily groups. Then a description of the set in question is given as union of certain quotient groups. The description of these quotients is the main point, and to do that the authors use as one of their main tools the work by W. D. Barcus and M. G. Barratt [Trans. Am. Math. Soc. 88, 57–74 (1958; Zbl 0095.16801)] and a generalization of it by J. W. Rutter [Topology 6, 379–403 (1967; Zbl 0152.21804)].
Reviewer: Daciberg Gonçalves (São Paulo)
MSC:
| 55P05 | Homotopy extension properties, cofibrations in algebraic topology |
| 55P10 | Homotopy equivalences in algebraic topology |
| 55P15 | Classification of homotopy type |
| 55Q05 | Homotopy groups, general; sets of homotopy classes |
| 55S37 | Classification of mappings in algebraic topology |
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