Portal de Periódicos da CAPES

Characteristic classes

1970; Duke University Press; Volume: 14; Issue: 3 Linguagem: Inglês

10.1215/ijm/1256053086

1945-6581

E. H. Connell,

Conner and Floyd have developed a theory of characteristic classes in gen- eralized cohomology [17].The purpose of this paper is to present an abstract development of their theory.The theory holds for real, complex, or quaterni- onic bundles, and the proofs for the three cases are essentially identical.The results hold for bundles over infinite complexes provided we consider only representable cohomology theories (called r-theories).The theory is based upon one theorem, the Thom-Dold isomorphism for r-theories and infinite complexes.A simple proof of this theorem is included.To outline the general development, consider only real vector bundles for the moment.If h is an r-theory, then any two of the following are equivalent" (i) There exists an element p f (RP) such that h** (RP) is an h** (pt) power series module over p.(ii) The Hopf bundle over RP is h-orientable.(iii) For each finite n, the Hopf bundle over RP is h-orientable.If h satisfies (i) above, h is said to be real orientable and p is said to be a real orientation for the cohomology theory h.Then the generalized Stiefel-Whit- ney classes exist, i.e. for each real bundle a over X, w(a) e h(X).Ifx h (RP) is zero when restricted to any RP', then x is zero, i.e. h (PR)has no phantom classes.Also h (BO) and h (BO) have no phantom classes.The groups h** (BO,) and h** (BO) are h** (pt) power series modules over the Stiefel-Whitney classes.Every real bundle is h-orientable.Any other orientation gives another set of SW classes , however w and will agree when restricted to the/-skeleton X.The set of all orientations corresponds to the set of all series of the form =i=p W a p aa pawhere a e h -(pt).Let KO (X) [X, BO X Z] be defined for infinite complexes.Then the do- main of the characteristic classes can be extended so that w KO (X) ---.h (X).Since real vector bundles are orientable for the ordinary theory H (-; Z), it follows that H* (PR", Z.) is a Z polynomial algebra, that the classical SW classes exist, and that H* (BO, Z) and H* (BO; Z.) are polynomial algebras over the SW classes.(Here it is unnecessary to pass to the direct product H**.)Now suppose h is an r-theory such that, each complex Hopf bundle .over CP'* is h-orientable.Then any orientation p .i (CP") determines Chern classes c K (X) -h (X), and the theory is analogous to the real case.The