On bordism groups of immersions. (English) Zbl 0546.57014
Summary: The bordism group of immersions of oriented n-manifolds into \({\mathbb{R}}^{n+k}\) is identified with the stable homotopy group \(\pi^ s_{n+k}(MSO(k)).\) We study these groups for n-2\(\leq k\leq n\), and discuss the behaviour of double points and their relation with the corresponding bordism groups of embeddings.
MSC:
| 57R42 | Immersions in differential topology |
| 57R90 | Other types of cobordism |
| 55Q10 | Stable homotopy groups |
| 55P42 | Stable homotopy theory, spectra |
References:
| [1] | Richard L. W. Brown, Immersions and embeddings up to cobordism, Canad. J. Math. 23 (1971), 1102 – 1115. · Zbl 0222.57018 · doi:10.4153/CJM-1971-116-8 |
| [2] | Ralph L. Cohen, Immersions of manifolds, Proc. Nat. Acad. Sci. U.S.A. 79 (1982), no. 10, 3390 – 3392. · Zbl 0495.57012 |
| [3] | David Ellis, Unstable bordism groups and isolated singularities, Trans. Amer. Math. Soc. 274 (1982), no. 2, 695 – 708. · Zbl 0517.57016 |
| [4] | Ulrich Koschorke, Vector fields and other vector bundle morphisms — a singularity approach, Lecture Notes in Mathematics, vol. 847, Springer, Berlin, 1981. · Zbl 0459.57016 |
| [5] | Mark E. Mahowald and Franklin P. Peterson, Secondary cohomology operations on the Thom class, Topology 2 (1963), 367 – 377. · Zbl 0166.19502 · doi:10.1016/0040-9383(63)90016-8 |
| [6] | W. S. Massey and F. P. Peterson, On the dual Stiefel-Whitney classes of a manifold, Bol. Soc. Mat. Mexicana (2) 8 (1963), 1 – 13. · Zbl 0121.18005 |
| [7] | C. Olk, Immersionen von Mannigfaltigkeiten in euklidische Räume, Dissertation, Siegen, 1980. · Zbl 0476.57014 |
| [8] | Hans A. Salomonsen, On the homotopy groups of Thom-complexes and unstable bordism, Proceedings of the Advanced Study Institute on Algebraic Topology (1970), Vol. III, Mat. Inst., Aarhus Univ., Aarhus, 1970, pp. 476 – 494. Various Publ. Ser., No. 13. · Zbl 0241.57016 |
| [9] | András Szűcs, Cobordism of maps with simplest singularities, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 223 – 244. |
| [10] | Emery Thomas, Embedding manifolds in Euclidean space, Osaka J. Math. 13 (1976), no. 1, 163 – 186. · Zbl 0328.57009 |
| [11] | C. T. C. Wall, Determination of the cobordism ring, Ann. of Math. (2) 72 (1960), 292 – 311. · Zbl 0097.38801 · doi:10.2307/1970136 |
| [12] | Robert Wells, Cobordism groups of immersions, Topology 5 (1966), 281 – 294. · Zbl 0145.20202 · doi:10.1016/0040-9383(66)90011-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
