Geodesic
Pontryagin manifolds
Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 441-447 
N. V. Ivanov. Pontryagin manifolds. Sbornik. Mathematics, Tome 36 (1980) no. 3, pp. 441-447. http://geodesic.mathdoc.fr/item/SM_1980_36_3_a9/
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Pontryagin manifold $P_{n,k}$ is the set of $(k+1)$-frames in $\mathbf R^n$ such that the dimension of the linear span of the vectors in the frame is no less than $k$. In the theory of Pontryagin classes these manifolds play a role analogous to that of the Stiefel manifolds in the theory of Stiefel–Whitney classes. The present paper examines the homotopy type of these manifolds. The results are then applied to study the connection between immersions and $k$-immersions. Bibliography: 8 titles.

[1] V. A. Rokhlin, “Vnutrennee opredelenie kharakteristicheskikh tsiklov Pontryagina”, DAN SSSR, 84:3 (1952), 449–452 | Zbl

[2] L. S. Pontryagin, “Kharakteristicheskie tsikly differentsiruemykh mnogoobrazii”, Matem. sb., 21(63) (1947), 233–284 | MR | Zbl

[3] L. S. Pontryagin, “Vektornye polya na mnogoobraziyakh”, Matem. sb., 24(66) (1949), 129–162 | MR | Zbl

[4] Wu Wen-Tsun, “Sur les classes caractéristiques des structures fibrées spheriques”, Actual Sci. Ind., 1183 (1952), 1–89

[5] A. Borel, “Sur les cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts”, Ann. Math., 57 (1953), 115–207 | DOI | MR | Zbl

[6] J.-P. Serre, “Croupes d'homotopie et classes de groupes abèliens”, Ann. Math., 58 (1953), 258–294 | DOI | MR | Zbl

[7] M. L. Gromov, “Stabilnye otobrazheniya sloenii v mnogoobraziya”, Izv. AN SSSR, seriya matem., 33 (1969), 707–734 | MR | Zbl

[8] N. V. Ivanov, “Mnogoobraziya Pontryagina”, VII Vsesoyuznaya topologicheskaya konferentsiya, tezisy dokladov i soobschenii, Minsk, 1977, 85