Zero-section Gysin and excess intersection for vector subbundles

Compatibility. Since $p^{\ast }$ is bijective with inverse $s_N^{!}$ by [L1], the displayed identities are equivalent after applying $p^{\ast }$ to the corresponding identities of bivariant operations: proper, flat and divisor compatibility of $s_N^{!}$ follows by applying $p^{\ast }$ and the corresponding push-pull or Cartier identities to each equality; all constructions are stable under base change by [L1] and [L2].

The excess formula. Consider the universal section of $p^{\ast }Q$ on $N$, namely the image of the vector coordinate under the composite $N\to Q$ (equivalently the section of $p^{\ast }Q$ whose value at a point is the class of the tautological vector); its zero scheme is exactly the subbundle $N^{\mathrm{\prime }}$. After a local splitting of $0\to N^{\mathrm{\prime }}\to N\to Q\to 0$ the section cuts $\mathrm{rank}Q$ independent fibre coordinates, so its zero scheme is regularly embedded of codimension $\mathrm{rank}Q$. For an integral $V\subseteq T$, write $N_V=p^{-1}(V)$ and $N_V^{\mathrm{\prime }}=p^{\mathrm{\prime }-1}(V)$. These are integral and pure-dimensional, of dimensions $\dim V+r$ and $\dim V+\mathrm{rank}{N}^{\mathrm{\prime }}$, with flat-pullback cycles $[N_V]=p^{\ast }[V]$ and $[N_V^{\mathrm{\prime }}]=p^{\mathrm{\prime }\ast }[V]$. On $N_V$ the restricted universal section again cuts independent fibre coordinates, so [L2] applies there. Pushing its section formula along $N_V\hookrightarrow N$ and using proper compatibility and flat naturality of Chern operators gives $a_{\ast }{p}^{\mathrm{\prime }\ast }[V]=c_{\mathrm{rank}Q}(p^{\ast }Q)\cap p^{\ast }[V]=p^{\ast }(c_{\mathrm{rank}Q}(Q)\cap [V])$. Applying $s_N^{!}$ and extending linearly gives $s_N^{!}{a}_{\ast }{p}^{\mathrm{\prime }\ast }\alpha =c_{\mathrm{rank}Q}(Q)\cap \alpha$ for all $\alpha$.

The zero-section case. Taking $N^{\mathrm{\prime }}=0$ and $Q=N$ gives the zero section $s:T\to N$ of $N$ and the identity $s_N^{!}{s}_{\ast }\alpha =c_r(N)\cap \alpha$; the local splittings used in step 1.2 verify regularity only and do not assert a global splitting of the sequence.