Punctured tubular neighborhoods and stable homotopy at infinity

TL;DR

This paper develops a comprehensive framework involving punctured tubular neighborhoods, orientation classes, and explicit models to connect stable homotopy theory at infinity with algebraic geometry, including quadratic forms and Chow-Witt groups.

Contribution

It introduces a new theory of orientation classes for line bundles over singular curves and refines the main theorem to incorporate these classes, linking homotopy and algebraic invariants.

Findings

Explicit models of punctured tubular neighborhoods

Quadratic Mumford matrices computed via Smith form

Enhanced understanding of orientation classes and trace computations

Abstract

In this revised version (August 2025), we add a survey of \infty-categorical (co)limits and a replacement lemma for higher functoriality (Lem. 1.4.5), a framework for explicit models of punctured tubular neighborhoods ({\S}3.4), and a new theory of orientation classes for line bundles and Thom spaces of virtual bundles over singular curves ({\S}5.1, 5.2). Building on this, we make explicit the normalization of the resulting isomorphisms, reformulate our main theorem (Th. 5.3.3) to incorporate orientation classes, and show how these choices yield quadratic Mumford matrices computed via Smith form over the Grothendieck-Witt ring of the base field. The appendices are expanded to give a better account of the notion of orientation classes, and to describe trace computations on Chow-Witt groups over possibly non-perfect fields. We warmly thank the referee for insightful comments that motivated us to make our approach much more precise and comprehensive.