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A clutching function is the equatorial transition map used in the clutching construction. For an oriented real vector bundle one may choose a fiber metric and orthonormal trivializations, so its clutching function takes values in the special orthogonal group.

A rank-$k$ complex vector bundle is a smooth map $\pi :E\to B$ whose fibers are $k$-dimensional complex vector spaces, with local diffeomorphisms ${\Phi }_i:{\pi }^{-1}(U_i)\to U_i\times {\mathbb{C}}^k$ commuting with projection and linear on every fiber. Define transition functions of a vector bundle byThey are smooth and obeyon the appropriate double/triple overlaps. The structure group of a vector bundle is a Lie subgroup $G\subset GL(k,\mathbb{C})$ in which a chosen set of transitions takes values. It records allowed changes of fiber frame; different trivializations may exhibit different reductions of that group.

Using the same cocycle, glue the disjoint union of $U_i\times G$ byThe cocycle makes this an equivalence relation and makes its local charts compatible. Right multiplication $(b,h{)}_i\cdot a=(b,ha{)}_i$ is well-defined because it commutes with the left gluing matrices. It is free and transitive on each fiber. These charts therefore construct the associated right principal bundle $P\to B$ with group $G$.

For the natural line realization, take the complex tautological line bundle $L=\{(\ell ,v):\ell \in {\mathbb{CP}}^1,v\in \ell \subset {\mathbb{C}}^2\}$. Its nonzero vectors identify with ${\mathbb{C}}^2\setminus \{0\}$ by $(\ell ,v)\mapsto v$, with projection $v\mapsto [v]$. On the two standard charts, use $\zeta =z_2/z_1$ and $\eta =z_1/z_2=1/\zeta$. The local frames are $e_0=(1,\zeta )$ and $e_1=(\eta ,1)$, so $e_1=e_0/\zeta$. Normalize them using the ambient Hermitian norm:If $v=t_0{u}_0=t_1{u}_1$, then $t_0=(|\zeta |/\zeta )t_1$. Thus the unitary transitions of the tautological line over the projective line areThese are smooth on the overlap $\zeta \ne 0$. A Hermitian metric on a smooth complex vector bundle generally permits precisely this unitary reduction; here the ambient norm supplies it explicitly.

The corresponding principal circle bundle consists of unit vectors in the tautological line. Sending $(\ell ,v)$ to $v$ identifies its total space with $S^3\subset {\mathbb{C}}^2$, and the right circle action is $v\cdot e^{i\theta }=e^{i\theta }v$. Identify ${\mathbb{CP}}^1$ with $S^2$ usingIts affine-coordinate inverse on the chart $Z\ne -1$ is $\zeta =(X-iY)/(1+Z)$; the other chart handles the missing pole. This gives a diffeomorphism, and the projection becomes the Hopf fibration

There is a distinction if the stated punctured-bundle identification is interpreted only as a projection-preserving diffeomorphism, rather than a fiber-linear identification. It does not determine whether the complex line is tautological or dual: for a nonzero $\varphi \in L_{\ell }^{\ast }$, the unique $v\in \ell$ with $\varphi (v)=1$ gives a smooth map $L^{\ast }\setminus 0\to L\setminus 0$. Locally it is inversion of a nonzero complex number. Thus punctured line bundles do not determine their duality sign.

For completeness, these are the only possible signs here. Complex line bundles over $S^2$ are described by the clutching construction with a loop in ${\mathbb{C}}^{\ast }$, reducible to $S^1$, and its integer winding number determines the bundle: a zero-degree ratio has a logarithm and can be removed by changes of trivializations extending over the two disks. For winding number $d$, the unit circle bundle has fundamental group $\mathbb{Z}/d\mathbb{Z}$, as the two solid-torus trivializations identify the common fiber generator and impose its $d$th power as the equatorial gluing relation. There is a fiberwise radial deformation retraction from the punctured line bundle onto its unit circle bundle. Since ${\mathbb{C}}^2\setminus 0$ retracts to simply connected $S^3$, the hypothesis forces $|d|=1$.

Consequently the allowed unitary transitions are the displayed $g_{01}$ or its reciprocal, according to the bundle's sign. For the dual, the same underlying Hopf projection has the inverse principal action $v\cdot e^{i\theta }=e^{-i\theta }v$. In either case the principal total space and projection are $S^3\to S^2$; the purely diffeomorphic hypothesis alone leaves the circle-action sign unspecified.

Let $D_{+}^n$ and $D_{-}^n$ be the two closed hemispheres of the sphere. Use $\gamma$ as the clutching function between the trivial bundles on these hemispheres. The clutching construction definesThe projection to $S^n$ is locally a vector bundle. In a collar of the equator, extend the transition function by keeping it constant in the normal coordinate; this gives explicit vector bundle trivializations across the seam. Since the transition matrices lie in the special orthogonal group, the standard fiber orientations and Euclidean inner products agree on overlaps. Thus this is an oriented rank-$k$ vector bundle with a fiber metric.

For $k\ge 2$, choose a base point $u_0$ on the equator. A constant change of the negative hemisphere frame replaces $\gamma$ by $\gamma (u_0{)}^{-1}\gamma$, so we may suppose $\gamma (u_0)=I$. With $e_k$ the last standard unit vector, define the section obstruction for a clutched vector bundleThe crucial point is that a nowhere-zero section can be normalized using the fiber metric. It is then represented by continuous maps ${\sigma }_{\pm }:D_{\pm }^n\to S^{k-1}$ satisfyingon the equator.

If $f_{\gamma }=0$, the extension-null-homotopy criterion for a sphere extends $u\mapsto \gamma (u)e_k$ to $D_{-}^n$. Take this extension as ${\sigma }_{-}$ and the constant map $e_k$ as ${\sigma }_{+}$. They glue to a nowhere-zero section.

Conversely, suppose such a section exists. Restricting a contraction of $D_{+}^n$ shows that ${\sigma }_{+}{|}_{S^{n-1}}$ is homotopic to a constant. The target sphere is path connected, so the constant can be moved to $e_k$. Multiplying this homotopy pointwise by $\gamma (u)$ gives a homotopy from $\gamma {\sigma }_{+}$ to $\gamma e_k$. The first of these maps equals ${\sigma }_{-}{|}_{S^{n-1}}$ and extends over $D_{-}^n$, hence is null-homotopic. So $\gamma e_k$ is null-homotopic too. A freely null-homotopic based sphere map represents zero in its homotopy group; any change of base point sends the zero class to zero. We have proved both directions:For $n=1$, the same argument reads as the extension criterion for a two-point boundary and uses the trivial set ${\pi }_0(S^{k-1})$ when $k\ge 2$. For $k=1$, the special orthogonal group $SO(1)$ is trivial, so the vector bundle is already a trivial vector bundle; the corresponding constant map to $S^0$ has zero obstruction. These cases require no higher homotopy group calculation.

If $k>n$, then $n-1<k-1$ and the permitted vanishing of lower homotopy groups of a sphere makes $f_{\gamma }=0$. Normalize the resulting nowhere-zero section to a unit section $\sigma$, and setLocal orthonormal frames show that these orthogonal complements form a rank-$(k-1)$ vector bundle. The explicit vector bundle isomorphismhas inverse $w\mapsto (w-⟨w,\sigma ⟩\sigma ,⟨w,\sigma ⟩)$. Orient $V^{\prime }$ so that a positive frame followed by $\sigma$ is positive in $V_{\gamma }$. This is splitting a trivial line from a nowhere-zero section and proves the oriented splittingFinally consider the special orthogonal sphere fibrationThe fiber over $e_{s+1}$ consists exactly of $\mathrm{diag}(B,1)$ with $B\in SO(s)$. Near any unit vector, the Gram-Schmidt process continuously completes it to an oriented orthonormal frame, giving local sections of $p$ and local product fiber bundle charts. In particular $p$ has the homotopy lifting property.

Let $\alpha :S^r\to SO(s+1)$ be based and suppose $r<s$. The composite $p\alpha$ is a based null-homotopic map by the given vanishing of ${\pi }_r(S^s)$. Lift a based null-homotopy of $p\alpha$, beginning at $\alpha$, to a homotopy $G:S^r\times [0,1]\to SO(s+1)$. The lift might initially move its base point $u_0$, but $G(u_0,t)$ always fixes $e_{s+1}$. Right multiplication by $G(u_0,t{)}^{-1}$ corrects this: the homotopystill covers the same sphere homotopy, remains based, and starts at $\alpha$. At its endpoint every matrix fixes $e_{s+1}$, so $\tilde{G}(-,1)$ lies in the included $SO(s)$. Thus every homotopy class in the target is represented by one from $SO(s)$:This proves surjectivity of special orthogonal stabilization below the sphere dimension. For $r=0$, this is the corresponding assertion about path components; the positive-rank special orthogonal groups are connected. The lifting argument also explains the zero term ${\pi }_r(S^s)$ in the long exact sequence of homotopy groups of a fibration.