Cobordism Theory Using Steenrod Operations

In this paper I give new proofs of the structure theorems for the unoriented cobordism ring [12] and the complex cobordism ring [8, 131. The proofs are elementary in the sense that no mention of the Steenrod algebra or Adams spectral sequence is made. In fact, the only result from homotopy theory which is used in an essential way is the Serre finiteness theorem in order to know that the complex cobordism group of a given dimension is finitely generated. The technique used here capitalizes on the fact that there are two rather different approaches to defining operations in the complex and unoriented cobordism generalized cohomology theories. The first proceeds via characteristic classes and leads to the LandweberNovikov operations [6, 91, while the second is the analog of the Steenrod power method due to tom Dieck [14]. Using the technique of “localization at the fixpoint set” (Atiyah-Segal [l], tom Dieck [15, 16]), it is possible to derive an equation expressing the Steenrod operation in terms of the LandweberNovikov operations in which the Steenrod operation is zero modulo terms of high filtration. One thereby obtains nontrivial relations involving the action of the Landweber-Novikov operations on the cobordism ring which can be used to show that the cobordism ring is generated by the coefficients of the formal group law expressing the behavior of cobordism Euler classes of line bundles under tensor product. From this, Lazard’s results [7] on formal group laws can be applied to neatly prove that the two cobordism rings are polynomial rings. The paper also contains two new results of interest. The main theorem of the paper shows that the reduced complex cobordism o*(X) of a finite complex is generated by its elements of positive degree as a module over the complex cobordism ring. By duality this implies that * Supported by the Alfred I’. Sloan Foundation, the National Science Foundation, and the Institute for Advanced Study.