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

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.