The $\mathbf{P}^1$-motivic Gysin map

TL;DR

This paper develops a new $ extbf{P}^1$-unstable motivic theory and constructs a Gysin map for regular immersions, unifying Gysin maps across various cohomology theories.

Contribution

It introduces a novel $ extbf{P}^1$-unstable motivic framework and provides a uniform construction of Gysin maps for multiple cohomology theories.

Findings

Constructed the Gysin map for regular immersions within the new theory.

Unified Gysin map construction for various cohomology theories.

Extended the Gysin map to Annala--Hoyois--Iwasa $ extbf{P}^1$-motivic spectra.

Abstract

We develop a $\mathbf{P}^1$-unstable non-$\mathbf{A}^1$-invariant theory of motivic spaces and spectra, and construct the Gysin map therein for regular immersions. This in particular gives the Gysin map in the Annala--Hoyois--Iwasa $\mathbf{P}^1$-motivic spectra, and thus gives a uniform construction for the Gysin maps of various cohomology theories.