Imbeddings, immersions, and characteristic classes of differentiable manifolds
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- Proc. Amer. Math. Soc. 69 (1978), 177-180 Request permission
Abstract:
Let $I_n^i$ be the set of $\bmod {\text { - }}2$ characteristic classes which are of dimension i, and they are zero for all n-dimensional smooth manifolds. Let $I_{n,k}^i$ be the set of i-dimensional $\bmod {\text { - }}2$ characteristic classes which are zero for all n-dimensional smooth manifolds which immerse in codimension k, (we are talking about normal characteristic classes). Let K be the (graded) ideal in ${H^ \ast }(BO,{Z_2})$ generated by ${w_{k + 1}},{w_{k + 2}}, \ldots$. Then if $i \leqslant (n + k)/2$, we have $I_{n,k}^i = I_n^i + {K^i}$. We have some related results for imbedded manifolds, and also for manifolds which immerse or imbed with an SO, U, SU, Spin, etc. structure on the normal bundle.References
- Martin Bendersky, Characteristic classes of $n$-manifolds immersing in $\textbf {R}^{n+k}$, Math. Scand. 31 (1972), 293–300 (1973). MR 324708, DOI 10.7146/math.scand.a-11436
- Edgar H. Brown Jr. and Franklin P. Peterson, Relations among characteristic classes. I, Topology 3 (1964), no. suppl, suppl. 1, 39–52. MR 163326, DOI 10.1016/0040-9383(64)90004-7
- Edgar H. Brown Jr. and Franklin P. Peterson, Relations among characteristics classes. II, Ann. of Math. (2) 81 (1965), 356–363. MR 176490, DOI 10.2307/1970620
- Richard L. W. Brown, Imbeddings, immersions, and cobordism of differentiable manifolds, Bull. Amer. Math. Soc. 76 (1970), 763–766. MR 259930, DOI 10.1090/S0002-9904-1970-12540-X
- Henri Cartan, Sur les groupes d’Eilenberg-Mac Lane. II, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 704–707 (French). MR 65161, DOI 10.1073/pnas.40.8.704
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242–276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
- R. Lashof, Poincaré duality and cobordism, Trans. Amer. Math. Soc. 109 (1963), 257–277. MR 156357, DOI 10.1090/S0002-9947-1963-0156357-4
- Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. Mathematical notes. MR 0248858
- René Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28 (1954), 17–86 (French). MR 61823, DOI 10.1007/BF02566923
- Robert Wells, Cobordism groups of immersions, Topology 5 (1966), 281–294. MR 196760, DOI 10.1016/0040-9383(66)90011-5
Additional Information
- © Copyright 1978 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 69 (1978), 177-180
- MSC: Primary 57D20; Secondary 55G36, 55G40, 57D40
- DOI: https://doi.org/10.1090/S0002-9939-1978-0488078-9
- MathSciNet review: 0488078