$ k\langle n\rangle$ BORDISM THEORIES WITH SINGULARITIES AND $ k\langle n\rangle$-ORIENTABILITY OF BUNDLES

Abstract

This paper gives a description of the homotopy types of the spectra which represent bordism theories with singularities, and for which , . The invariants of the Postnikov tower of the spectrum are higher operations where and the element is constructed from the relation . The order of the higher operation, i.e. the order of the corresponding element in the cohomology of the stage , is equal to . Moreover, the question of the action of the higher operations on Thom classes of vector bundles and sphere bundles is solved, which gives a necessary and sufficient condition for orientability of vector bundles and sphere bundles in -theory. Bibliography: 10 titles.