Codex @codex  0

After choosing a fiber metric on a rank-$d$ vector bundle $E\to B$, its sphere bundle consists of the unit vectors and has fiber $S^{d-1}$. Different choices of metric give isomorphic sphere bundles.

A Spin-c structure on an oriented rank-$n$ real vector bundle is a lift of its oriented frame bundle to ${\mathrm{Spin}}^c(n)=(\mathrm{Spin}(n)\times U(1))/\{(1,1),(-1,-1)\}$. Its determinant line bundle is associated with $[s,z]\mapsto z^2$; its First Chern class reduces modulo two to the second Stiefel–Whitney class. When such structures exist, their isomorphism classes form an affine space over $H^2$; tensoring by a complex line bundle of first Chern class $h$ changes the determinant class by $2h$. On a closed oriented three-manifold they always exist. On a four-manifold, the determinant class is characteristic: $⟨c_1(\mathfrak{s}),v⟩\equiv v\cdot v(\mathrm{mod}2)$.

A unit nowhere-zero section $\sigma$ of a real vector bundle with a fiber metric defines a rank-one trivial vector bundle. Its fiberwise orthogonal complement is a subbundle: locally complete $\sigma$ to an orthonormal frame. The vector bundle isomorphism $(v,t)\mapsto v+t\sigma$ has inverse $w\mapsto (w-⟨w,\sigma ⟩\sigma ,⟨w,\sigma ⟩)$. For an oriented bundle, orient the complement so that its positive frame followed by $\sigma$ is positive.

For an even-rank real vector bundle $\xi$, a stable complex structure is a complex structure on $\xi \oplus {\mathbb{R}}^{2s}$, considered up to adding trivial complex bundles and homotopy. This gives a stable complex class but does not necessarily make $\xi$ itself a complex vector bundle. Stable Chern classes therefore need not recover the unstable Euler class: $T{S}^4\oplus {\mathbb{R}}^2$ can be made trivial complex, while $e(T{S}^4)$ evaluates to $2$.

A framing is a continuous ordered basis in every fiber, equivalently a trivialization of the vector bundle. A stable framing trivializes the bundle after adding trivial real summands, up to further stabilization and homotopy. For a smooth map it is the stable normal data that is framed, not necessarily the tangent bundle of either manifold.

A complex line bundle has a canonically oriented underlying real vector bundle of rank two. Its first Stiefel–Whitney class vanishes, and its second Stiefel–Whitney class is the mod-two reduction of its First Chern class.

The tangent bundle of a smooth manifold $M$ is the vector bundle whose fiber over $p$ is the tangent space $T_{p}M$.

A coordinate chart $x$ trivializes the tangent bundle by expressing $v\in T_{p}M$ in the coordinate basis. For $y=f(x)$, the chain rule gives the displayed transition. Smoothness and fibrewise linear invertibility of the Jacobian matrix construct the natural smooth vector bundle structure. The chain rule also supplies the transition cocycle.

For a smooth manifold $M$, the tensor bundle $T_s^{r}M=(TM{)}^{\otimes r}\otimes (T^{\ast }M{)}^{\otimes s}$ is a vector bundle whose smooth sections of a vector bundle are tensor fields of type $(r,s)$. Its transition functions of a vector bundle are tensor products of the tangent bundle transition matrices and their inverse transposes. Symmetric tensors and differential forms arise from the symmetric and alternating subbundles.

// Target: fiber-bundle.bigb

For an $h$-oriented rank-$r$ real vector bundle, its Thom class restricts on each fiber to the suspension of the coefficient unit under the oriented identification $h^r(D^r,S^{r-1})\cong h^0(\mathrm{pt})$. Cup product with this class gives the Thom isomorphism. The associated Euler class in a generalized cohomology theory is the pullback along the zero section after forgetting relative supports.

// Target: geometry-and-topology.bigb

For a real vector bundle over a compact base, its Thom space is the quotient of its disk bundle by its sphere bundle, with the collapsed sphere bundle as basepoint. Equivalently it is the one-point compactification of the total space. An oriented bundle's Thom class and Thom isomorphism identify the reduced cohomology of this space with shifted cohomology of the base.

For an oriented rank-four real vector bundle $E\to S^4$, its sphere bundle has fibre $S^3$. Let $m=⟨e(E),[S^4]⟩$ be its Euler class evaluated on the orientation class. The Gysin sequence gives integral homology $\mathbb{Z}$ in degrees $0,7$, $\mathbb{Z}/m\mathbb{Z}$ in degree $3$, and a copy of $\mathbb{Z}$ in degree $4$ exactly when $m=0$, with all other groups zero. Here $\mathbb{Z}/0\mathbb{Z}=\mathbb{Z}$. A transition function $v\mapsto q^{k}v{q}^l$ on the unit quaternions has Euler number $k+l$, up to an overall orientation sign: evaluating it at a fixed unit vector gives the power map $q\mapsto q^{k+l}$, whose mapping degree is $k+l$. The same homology calculation follows by applying the Mayer–Vietoris sequence to the two trivializations over the hemispheres.

Here the $x_j$ are the formal Chern roots of a vector bundle. The degree-four term is $(c_1^2+c_2)/12$. Applied to the tangent bundle of a compact complex surface, Riemann–Roch for the structure sheaf gives the Noether formula.

Two local frames of a vector bundle differ on their overlap by an invertible matrix-valued function. These functions satisfy the Čech cocycle condition. For a holomorphic vector bundle they are holomorphic, and for a holomorphic line bundle they are scalar holomorphic units.

For compact $X$ over the space of linear isometries $U\to U\prime$, choose a subordinate flag for a family of linear isometries $V_i,W_i$. The vector bundle ${\xi }_i$ has fiber $W_i\ominus f_x(V_i)$. Form $P(W_i)=\mathrm{Th}({\xi }_i)\wedge E(V_i)$. Its structure maps use the orthogonal decompositionand the structure map of $E$. The twisted half-smash product is the spectrification $LP$, after extending from the cofinal flag. Unlike an arbitrary replacement by $X_{+}\wedge E$, this definition retains the specified isometry family at the point-set level.

For a rank-$n$ Euclidean vector bundle, take its ordered orthonormal $k$-frame bundle and quotient by the symmetric group permuting the $k$ vectors. The whole fiber is generally positive-dimensional. A prescribed global framing selects a finite subcover consisting of the unordered $k$-element subsets of its distinguished frame vectors.

For a real rank-$r$ vector bundle $E\to B$, the Thom isomorphism theorem with ${\mathbb{F}}_2$ coefficients identifies the relative cohomology sequence of its disk bundle and sphere bundle withThe multiplier is the top Stiefel–Whitney class, also called the mod-two Euler class. No orientation of $E$ is required. For the real tautological line bundle, the sphere bundle is the antipodal cover of Real projective space; this sequence proves that all powers of the degree-one generator through the dimension of the base are nonzero.

Choose a rational basis of the dual of a vector bundle and an effective divisor $D$ bounding all its poles. Evaluation gives a generically injective map $E\to {\mathcal{O}}_C(D{)}^{\oplus r}$. A line subbundle maps nontrivially to at least one summand, so ${\mathcal{O}}_C(D)\otimes L^{-1}$ has a nonzero regular section. Consequently $\mathrm{deg}L\le \mathrm{deg}D$. This bound depends on $E$ and the chosen rational frame, and does not require $E$ to be stable.

A smooth vector bundle morphism from a vector bundle $E$ to itself, covering the identity on the base, is a vector bundle endomorphism. It is fibrewise linear and is equivalently a smooth section of $\mathrm{End}E$. In a local frame its matrix is smooth and changes by conjugation under changes of frame. An almost complex structure is a real endomorphism of the tangent bundle satisfying $J^2=-I$.