# Cobordism-framed correspondences and the Milnor K-theory

## TL;DR

This paper establishes an isomorphism between the 0th cohomology of cobordism-framed correspondences and Milnor K-groups, advancing the understanding of algebraic K-theory and cobordism.

## Contribution

It computes the 0th cohomology of cobordism-framed correspondences and proves its isomorphism to Milnor K-groups, extending previous results for framed correspondences.

## Key findings

- Isomorphism between cohomology of cobordism-framed correspondences and Milnor K-groups.
- Potential foundation for computing homotopy groups of MGL spectrum.
- Extension of Neshitov's results to cobordism-framed correspondences.

## Abstract

In this work, we compute the $0$th cohomology group of a complex of groups of cobordism-framed correspondences, and prove the isomorphism to Milnor $K$-groups. An analogous result for common framed correspondences has been proved by A. Neshitov in his paper "Framed correspondences and the Milnor---Witt $K$-theory". Neshitov's result is, at the same time, a computation of the homotopy groups $\pi_{i,i}(S^0)(Spec(k)).$ This work could be used in the future as basis for computing homotopy groups $\pi_{i,i}(MGL_{\bullet})(Spec(k))$ of the spectrum $MGL_{\bullet}.$