$k\langle n\rangle$ bordism theories with singularities and $k\langle n\rangle$-orientability of bundles

A. V. Khokhlov

Sbornik. Mathematics, Tome 51 (1985) no. 1, pp. 255-266

Résumé

This paper gives a description of the homotopy types of the spectra $k\langle n\rangle$ which represent bordism theories with singularities, and for which $\pi_*(k\langle n\rangle)=Z_{(p)}[t]$, $\dim t=2p^n-2$. The invariants of the Postnikov tower of the spectrum $k\langle n\rangle$ are higher operations $\widetilde Q_n^{(s)}$ where $\widetilde Q_n^{(0)}\in HZ_{(p)}*(HZ_{(p)})$ and the element $\widetilde Q_n^{(s+1)}$ is constructed from the relation $\widetilde Q_n^{(0)}\widetilde Q_n^{(s)}=0$. The order of the higher operation, i.e. the order of the corresponding element $\alpha_s$ in the cohomology of the stage $k^{s-1}\langle n\rangle$, is equal to $p^s$. Moreover, the question of the action of the higher operations $\widetilde Q_n^{(s)}$ 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 $k\langle n\rangle$-theory. Bibliography: 10 titles.