Let N and P be smooth closed manifolds of dimensions n and p respectively. Given a Thom-Boardman symbol I, a smooth map f : N → P is called an Ω -regular map if and only if the Thom-Boardman symbol of each singular point of f is not greater than I in the lexicographic order. We will represent the group of all cobordism classes of Ω -regular maps of n-dimensional closed manifolds into P in terms...

The Hopkins-Mahowald theorem realizes the Eilenberg-Maclane spectra $H\mathbb F_p$ as Thom for all primes $p\in\mathbb N_{>0}$. In this article, we record a known proof of generalization theorem, realizing $Hk$ perfect rings $k$, and provide further by $HR$ perfectoid $R$. We also discuss even generalizations to prisms $(A,I)$ indicates how adapt our proofs Breuil-Kisin case.

We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and vector bundles.