Thom isomorphisms glue over two trivializing opens

Put $D_T=D(\xi ){\mathrm{\mid }}_T$ and $S_T=S(\xi ){\mathrm{\mid }}_T$. Let $C_{\ast }^{\mathcal{U}}(D_{U\cup V},S_{U\cup V})$ be the quotient of $C_{\ast }(D_U)+C_{\ast }(D_V)$ by its sphere subcomplex. The small-chain subdivision of [F3] makes its inclusion in the ordinary relative chain complex a chain-homotopy equivalence. There is a degreewise split exact chain sequence $0\to C_{\ast }(D_W,S_W)\to C_{\ast }(D_U,S_U)\oplus C_{\ast }(D_V,S_V)\to C_{\ast }^{\mathcal{U}}(D_{U\cup V},S_{U\cup V})\to 0,$ where the last map is addition and the first is the signed pair of inclusions. Dualizing this split sequence gives a termwise exact cochain sequence whose first term is the small relative cochain complex and whose other maps are restriction and restriction-difference. Transporting its cohomology through the small-chain equivalence gives the ordinary relative Mayer–Vietoris sequence. The usual kernel/image chase in [F2] fixes its connecting map and signs.

Compatibility says $(u_U,u_V)$ lies in the kernel of the relative difference map. Exactness in step 1.1 supplies $u\in H^n(D_{U\cup V},S_{U\cup V};R)$ restricting to both local classes. Each fiber lies in at least one open set, so these restrictions show that $u$ is normalized.

For every $k$, place the base Mayer–Vietoris sequence from [F2] over the relative sequence of step 1.1 shifted by $n$, and use cup product with $u$ vertically. Restrictions commute by relative naturality in [F4]. For the Mayer–Vietoris connector, choose a cochain on one member lifting a difference cocycle. The lower connector is represented by its coboundary. The Leibniz rule in [F4], together with $\delta u=0$, says that the coboundary of the lifted cochain cupped with $u$ is its coboundary cupped with $u$, up to the fixed degree sign. Thus the connector square commutes with that unit sign by a direct calculation in the termwise split small-cochain sequences of step 1.1. The four outer vertical maps at the $U$, $V$, and $W$ terms are the stipulated local Thom isomorphisms. Hence [F5] makes the middle map $H^k(U\cup V;R)\to H^{k+n}(D_{U\cup V},S_{U\cup V};R)$ an isomorphism.

If $u^{\mathrm{\prime }}$ is another normalized gluing class, the isomorphism in step 3.1 writes $u-u^{\mathrm{\prime }}={\pi }^{\ast }a⌣u$ for a unique $a\in H^0(U\cup V;R)$. Restriction to any fiber evaluates $a$ at its basepoint times the orientation generator; normalization makes this zero, so $a$ vanishes on every component and $u=u^{\mathrm{\prime }}$. If $U$, $V$, or $W$ is empty, the sequence reduces to an identity or disjoint finite additivity and the same argument applies. Rank zero, the zero ring, a one-set cover, zero classes, both difference-map endpoints, and every graded connector sign are included. Exactness supplies one class for one compatible pair; it does not choose an indexed family, so no AC is used.