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Killing rational characteristic classes by surgery
Published online by Cambridge University Press: 18 May 2009
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Let fr:Xr → BO(r) be a sequence of fibrations with maps gr:Xr → Xr+1 such that the usual diagram commutes. For such a situation R. Lashof defines the concept of an X-structure on manifolds (see [3]), and proves a Thom-isomorphism for the cobordism groups of such manifolds. Let n, m be positive integers which are fixed throughout this paper. If r is very big in comparison with n + m then Xr is a simply connected CW-complex and the map (gr)*:H*(Xr; Q)→ H*(Xr+l; Q) is an isomorphism up to dimension n. Let γ be the pull-back over Xr of the universal r-linear bundle (which is, of course, a bundle over BO(r)). If r is very big in comparison with n + m, then we put X = Xr, and we assume that γ is orientable and oriented. The elements of H*(X; Q) of dimension not greater than n, will be called rational universal X-characteristic classes. It is well-known that many of the usual classes of manifolds may be described in terms of X-structures, (e.g. SO, SU, Spin-manifolds etc.).
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- Copyright © Glasgow Mathematical Journal Trust 1980
References
REFERENCES
1.Berstein, I., Homotopy mod C of spaces of category 2, Comment. Math. Helv. 35 (1961), 9–14.CrossRefGoogle Scholar
3.Lashof, R., Poincaré duality and cobordism, Trans. Amer. Math. Soc. 109 (1963), 257–277.Google Scholar
4.Mosher, R. and Tangora, M., Cohomology operations and applications in homotopy theory (Harper and Row, 1968).Google Scholar
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