Gysin pushforward for an oriented zero section

Assume AC. Let $\xi$ be an $R$-oriented rank-$n$ numerable vector bundle over a CW complex, or over a paracompact Hausdorff base of CW type. For its zero section define $s_{!}:H^k(B;R)⟶H^{k+n}(D(\xi );R);s_{!}(a)=j^{\ast }({\pi }^{\ast }a⌣u_{\xi }),$ where $j^{\ast }$ is the relative-to-absolute map. This is the Gysin pushforward of the oriented zero section.

The radial homotopy $v\mapsto (1-t)v$ retracts $D(\xi )$ to $s(B)$, so $s^{\ast }$ identifies the target with $H^{k+n}(B;R)$. Under that identification, cup-product naturality and the Euler definition give $s^{\ast }{s}_{!}(a)=a⌣s^{\ast }{j}^{\ast }{u}_{\xi }=a⌣e_{\mathrm{T}\mathrm{h}}(\xi )\mathrm{.}$

For rank zero, $D(\xi )=B$, $S(\xi )=\mathrm{\varnothing }$, and the supplied Thom class is its orientation unit $o\in H^0(B;R)$. Thus $s_{!}(a)=a⌣o$ and $e=o$, by the displayed composite. This is an isomorphism with inverse multiplication by $o^{-1}$, and is the identity for the standard unit orientation $o=1$. Empty bases and the zero ring give the unique zero maps. At $k=0$ the unit maps to $j^{\ast }u$; negative-degree sources are zero. Both radial endpoints, the zero section, identity bundle maps, and zero inputs are included. The displayed construction is choice-free after $u_{\xi }$ is supplied; AC is inherited only from the general existence theorem.