The Euler class construction remains owned by algebraic topology

This page proves zero-set and self-intersection applications of characteristic classes. Algebraic topology owns the normalized Thom class Thom class by fiberwise normalization, its existence and Thom isomorphism Thom isomorphism for oriented vector bundles, and the Euler class $e(E)=s_0^{\ast }{j}^{\ast }{u}_E$ Thom-defined Euler class of an oriented vector bundle, Euler class by zero-section pullback of the Thom class. Its naturality and ordered Whitney product are proved in Naturality, orientation sign, and Whitney product for Euler classes (with standard unit orientations on rank-zero inputs); the equality of the canonical mod 2 Euler class with the top Stiefel-Whitney class is proved in The mod-two Euler class is the top Stiefel–Whitney class. Each result retains its stated base, orientation and choice hypotheses. DT-12 cites these constructions and does not introduce competing ones. Conversely, zero-locus duality The zero locus of a transverse section represents the Euler dual and the normal-bundle self-intersection formula The self-intersection number is the Euler number of the normal bundle are DT-owned interfaces for the later Euler/index and immersion/embedding pairs.