The 𝐏¹-motivic Gysin map

Recently, Annala, Hoyois, and Iwasa have defined and studied the 𝐏¹-homotopy theory, a generalization of 𝐀¹-homotopy theory that does not require 𝐀¹ to be contractible, but only requires pointed 𝐏¹ to be invertible. This makes it applicable to non-𝐀¹-invariant cohomology theories such as Hodge, de Rham, and prismatic.] I will recall basic facts in their theory, and construct the 𝐏¹-motivic Gysin map[, thus giving a uniform construction for the Gysin maps of the above cohomology theories that are automatically functorial. If time permits, I will also explain some applications such as Atiyah duality and Steinberg relation in 𝐏¹-homotopy theory.