Normalized clutching data for bundles over X×S²

Let $D_{+}^2$ and $D_{-}^2$ be the closed hemispheres. Each inclusion $X\times \{0\}\hookrightarrow X\times D_{\pm }^2$ is a homotopy inverse to projection. By [F1], there are bundles $E_{\pm }$ on $X$ and isomorphisms $V{\mathrm{\mid }}_{X\times D_{\pm }^2}\cong {\mathrm{pr}}_X^{\ast }{E}_{\pm }$. In these trivializations, $V$ is obtained by an equatorial isomorphism $f(x,z):(E_{+}{)}_x\to (E_{-}{)}_x$.

At $z=1$, $f(x,1)$ is an isomorphism $E_{+}\to E_{-}$. Identify $E_{-}$ with $E=E_{+}$ by $f(x,1{)}^{-1}$. In the fixed coefficient convention the transition becomes $f(x,1{)}^{-1}f(x,z)$, which equals the identity at $z=1$. Thus $V=[E,f]$ with normalized $f$.

If $h_{\pm }$ and $h_{\pm }^{\mathrm{\prime }}$ are two normalized hemisphere trivializations of the same bundle, their ratios are maps $g_{\pm }:X\times D_{\pm }^2\to \mathrm{Aut}(E)$ with $g_{\pm }(x,1)=I$. The straight contraction of each disk to $1$ fixes $1$, so composing $g_{\pm }$ with it gives homotopies to the identity through maps still equal to $I$ at $1$. Applying these changing gauges to the equatorial transition gives a homotopy between the two normalized clutching maps.

For a virtual class, [F3] complements its negative bundle into a finite trivial bundle and then applies steps 1.1–3.1 to the resulting actual bundle; this is the optional stabilization in the statement.

A normalized homotopy $f_t$ glues, by [F2], a bundle on $X\times S^2\times I$. Its endpoint restrictions are isomorphic by [F1]. The normalization keeps the chosen common bundle and basepoint frame fixed, and adding trivial summands before the homotopy gives the same conclusion for stabilized data.