On the reduction of complex bordism to unoriented bordism

Johnson, David Copeland

doi:10.1090/S0002-9939-1973-0322853-1

On the reduction of complex bordism to unoriented bordism
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Proc. Amer. Math. Soc. 39 (1973), 417-420 Request permission

Abstract:

The image of the natural transformation from the complex bordism of a CW complex $X,M{U_ \ast }(X)$, to its unoriented bordism, ${N_ \ast }(X)$, is contained in a subgroup identified with ${H_ \ast }(X;Z) \otimes {({N_ \ast })^2}$. A characterization is given for the CW complexes for which the image and the subgroup coincide.

References

Theodor Bröcker and Tammo tom Dieck, Kobordismentheorie, Lecture Notes in Mathematics, Vol. 178, Springer-Verlag, Berlin-New York, 1970 (German). MR 0275446
P. E. Conner and Larry Smith, On the complex bordism of finite complexes, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 117–221. MR 267571
Peter S. Landweber, Complex bordism of classifying spaces, Proc. Amer. Math. Soc. 27 (1971), 175–179. MR 268885, DOI 10.1090/S0002-9939-1971-0268885-1
J. Milnor, On the Stiefel-Whitney numbers of complex manifolds and of spin manifolds, Topology 3 (1965), 223–230. MR 180977, DOI 10.1016/0040-9383(65)90055-8
George E. Mitchell, The image of ${\mathfrak {U}}_\ast (X)\rightarrow {\mathfrak {N}}_\ast (X)$, Proc. Amer. Math. Soc. 26 (1970), 505–508. MR 278309, DOI 10.1090/S0002-9939-1970-0278309-5
Daniel Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971), 29–56 (1971). MR 290382, DOI 10.1016/0001-8708(71)90041-7