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We initiate a study of punctured tubular neighborhoods and homotopy theory atinfinity in motivic settings. We use the six functors formalism to give anintrinsic definition of the stable motivic homotopy type at infinity of analgebraic variety. Our main computational tools include cdh-descent for normalcrossing divisors, Euler classes, Gysin maps, and homotopy purity. Under$\ell$-adic realization, the motive at infinity recovers a formula forvanishing cycles due to Rapoport-Zink; similar results hold for Steenbrink'slimiting Hodge structures and Wildeshaus' boundary motives. Under thetopological Betti realization, the stable motivic homotopy type at infinity ofan algebraic variety recovers the singular complex at infinity of thecorresponding topological space. We coin the notion of homotopically smoothmorphisms with respect to a motivic $\infty$-category and use it to show ageneralization to virtual vector bundles of Morel-Voevodsky's purity theorem,which yields an escalated form of Atiyah duality with compact support. Further,we study a quadratic refinement of intersection degrees, taking values inmotivic cohomotopy groups. For relative surfaces, we show the stable motivichomotopy type at infinity witnesses a quadratic version of Mumford's plumbingconstruction for smooth complex algebraic surfaces. Our construction andcomputation of stable motivic links of Du Val singularities on normal surfacesis expressed entirely in terms of Dynkin diagrams. In characteristic p>0, thisimproves Artin's analysis on Du Val singularities through \'etale localfundamental groups. The main results in the paper are also valid for$\ell$-adic sheaves, mixed Hodge modules, and more generally motivic$\infty$-categories.

Punctured tubular neighborhoods and stable homotopy at infinity