📝 SummarXiv

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.