List of Geometric Bundles and Invariants

Introduction

This article lists the bundles of the corpus — the fibre, vector and principal bundles — the connections carried on them, and the characteristic classes and characteristic numbers that measure each bundle. Every entry points to the article that introduces the object, and the article introduces nothing and proves nothing.

The list follows the chain that the two articles of the theme set out. A fibre bundle is a local product with transition functions; a principal bundle carries the structure group and the associated bundles are built from it; a connection on a bundle is the rule of parallel transport, and its curvature is the local measure of the failure of the transport to commute; the invariant polynomials of the curvature give the characteristic classes, and the evaluation of those classes on a manifold gives the characteristic numbers that the index theorem computes. The invariants of the list are the classes and numbers that are independent of the connection, and the classifying spaces and K-theory are the functors in which the classes are naturally indexed.

The article records examples and non-examples side by side. Beside the bundles and their invariants it lists the structures that are not bundles, the bundles that carry no Euler class because they are not orientable, and the characteristic classes that vanish for a flat connection, each with the failure named and the article that records it.

Fibre, Vector and Principal Bundles

A fibre bundle is a map that is locally a product, and the structure of the bundle is encoded in its transition functions and in its structure group.

Object	The property it has	Introduced in
a fibre bundle $\pi : E \to B$	locally a product $U\times F$, with fibre $F$	Fibre Bundles, Connections and Curvature
the transition functions $g_{\alpha\beta}$	the clutching data, with $g_{\alpha\beta}g_{\beta\gamma} = g_{\alpha\gamma}$	Fibre Bundles, Connections and Curvature
a principal $G$-bundle $P \to M$	a bundle with a free transitive right action of the structure group $G$	Fibre Bundles, Connections and Curvature
the associated bundle $P\times_G F$	the bundle with fibre $F$ built from a $G$-space and a principal bundle	Fibre Bundles, Connections and Curvature
the frame bundle of $M$	the principal bundle of the frames, whose associated bundles are the tensor bundles	Fibre Bundles, Connections and Curvature
the tangent and cotangent bundles $TM$, $T^*M$	the vector bundles of the tangent and cotangent spaces	Smooth Manifolds and Differential Geometry
a vector bundle $E \to B$	a fibre bundle with a vector space as fibre and linear transitions	Fibre Bundles, Connections and Curvature; Characteristic Classes
the section space $\Gamma(E)$	the smooth sections of a bundle; the fields of the theory	Fibre Bundles, Connections and Curvature
the tautological bundle $\gamma_k$ on a Grassmannian	the rank-$k$ bundle whose fibre is the subspace itself	Grassmannians and Stiefel Manifolds
a line bundle and its class in $H^2(B;\mathbb{Z})$	the isomorphism classes of complex line bundles	Characteristic Classes
the canonical bundle $K_M$	the top exterior power of the holomorphic cotangent bundle	Kähler Geometry
the normal bundle and the tubular neighbourhood	the bundle of the normal directions of a submanifold, and its disc bundle	Differential Topology
the Hopf fibration $S^3 \to S^2$	the principal $U(1)$-bundle of the three-sphere over the sphere	Spherical Geometry, §The Three-Sphere and the Hopf Fibration; Homotopy Groups and Fibrations

The transition functions are the complete cocycle data of the bundle, and the principal bundle is the frame bundle of a vector bundle, so that the geometry of a vector bundle and the geometry of its frame bundle carry the same information. The associated-bundle construction produces from a representation of the structure group the tensor, spinor and density bundles that the corpus uses on manifolds.

Connections and Curvature

A connection is a rule of parallel transport, given locally by a connection form, and its curvature measures the failure of the transport to commute.

Object	The property it has	Introduced in
a connection $\nabla$	a derivation of the sections, $\nabla(fs) = df\otimes s + f\nabla s$	Fibre Bundles, Connections and Curvature
the connection form in a frame $\omega = (\omega^i_{\ j})$	$\nabla = d+\omega$, matrix-valued in the frame	Fibre Bundles, Connections and Curvature
the change of frame $\omega' = g^{-1}\omega g + g^{-1}dg$	the transformation of the connection, the gauge transformation rule	Fibre Bundles, Connections and Curvature
a connection form on a principal bundle $\omega \in \Omega^1(P;\mathrm{G})$	$\omega(A^{\#}) = A$ and $R_g^*\omega = \operatorname{Ad}(g^{-1})\omega$	Fibre Bundles, Connections and Curvature
the horizontal subspace $H_pP = \ker\omega_p$	the complement to the vertical, defining the transport	Fibre Bundles, Connections and Curvature
the curvature $R = d\omega + \omega\wedge\omega$	the $\operatorname{End}(E)$-valued $2$-form measuring noncommutation	Fibre Bundles, Connections and Curvature
the curvature of a principal connection $\Omega = d\omega + \frac12[\omega,\omega]$	the $\mathrm{G}$-valued curvature form	Fibre Bundles, Connections and Curvature
the Bianchi identity $d^\nabla R = 0$	the differential condition the curvature satisfies	Fibre Bundles, Connections and Curvature
the holonomy group of a connection	the group of the parallel transports around loops	Fibre Bundles, Connections and Curvature
a flat connection	the connection with $R = 0$, locally trivial or a Maurer–Cartan form	Fibre Bundles, Connections and Curvature
a gauge transformation	a change of frame, acting on the connection by the transformation rule	Fibre Bundles, Connections and Curvature
the Levi-Civita connection	the metric, torsion-free connection of the tangent bundle	Riemannian Geometry

The curvature is tensorial, so its value at a point depends only on the vectors and not on the extension of the sections, and the Bianchi identity makes it the closed form of the connection. A flat connection has a global system of parallel sections after the passage to the universal cover, and the holonomy group is the algebraic invariant of a connection that the characteristic classes do not see; the Levi-Civita connection is the case in which the connection is determined by a metric.

Characteristic Classes

The invariant polynomials of the curvature are closed and their de Rham classes are independent of the connection, and they are the characteristic classes of the bundle.

Class	The property it measures	Introduced in
the Chern–Weil homomorphism	the map $S^\bullet(\mathrm{G}^*)^G \to H^{\mathrm{ev}}_{dR}(M)$ from invariant polynomials to cohomology	Fibre Bundles, Connections and Curvature
the first Chern class $c_1(L)$	the class of $\frac{i}{2\pi}\mathcal{F}$, the curvature of a line bundle	Fibre Bundles, Connections and Curvature
the Euler class $e(E)$	the class of a real oriented bundle of even rank, the obstruction to a nowhere-zero section	Characteristic Classes
the Stiefel–Whitney classes $w_i(E) \in H^i(B;\mathbb{Z}/2)$	the $\mathbb{Z}/2$ classes; $w_1$ is orientability and $w_2$ is the spin obstruction	Characteristic Classes
the Chern classes $c_i(E) \in H^{2i}(B;\mathbb{Z})$	the integral classes of a complex bundle, with $c(E\oplus F) = c(E)c(F)$	Characteristic Classes
the Pontryagin classes $p_i(E)$	the real classes, $p_i(E) = (-1)^ic_{2i}(E\otimes\mathbb{C})$	Characteristic Classes
the Chern roots and the splitting principle	the formal factorisation $c(E) = \prod_i(1+x_i)$	Characteristic Classes
the Chern character $\operatorname{ch}(E)$	the rational class with $K^0(B)\otimes\mathbb{Q} \cong H^{\mathrm{ev}}(B;\mathbb{Q})$	Characteristic Classes
the relations $p_i \equiv w_{2i}^2$ and $e \equiv w_n \pmod 2$	the compatibility of the integral and $\mathbb{Z}/2$ classes	Characteristic Classes
the Hirzebruch $L$-polynomial $L_k(p_1,\ldots,p_k)$	the multiplicative polynomial $\sum_kL_k = \prod_j\frac{x_j}{\tanh x_j}$	Characteristic Classes
the Todd class	the multiplicative class of the index theorem for the Dolbeault complex	The Atiyah–Singer Index Theorem and K-Theory
the $\hat A$-genus	the multiplicative class of the Dirac operator	Spin Geometry; The Atiyah–Singer Index Theorem and K-Theory

The characteristic classes are the cohomological invariants of a bundle, they are natural under pullback, and they are computable from any connection by the invariant polynomials of its curvature. The splitting principle reduces the computation to the case of a sum of line bundles, and the multiplicative sequences — the $L$-polynomial, the Todd class and the $\hat A$-genus — are the generating series in which the classical index theorems are stated.

Characteristic Numbers and the Index Theorem

A characteristic number is the evaluation of a characteristic class of the tangent bundle on the fundamental class, and the index theorem identifies it with the index of an elliptic operator.

Object	The property it has	Introduced in
a characteristic number $\langle\kappa(TM),[M]\rangle$	the evaluation of a class of $TM$ on the fundamental class of a closed manifold	Characteristic Classes
the Gauss–Bonnet theorem $\int_Me(TM) = \chi(M)$	the Euler class integrates to the Euler characteristic	Fibre Bundles, Connections and Curvature; Riemannian Geometry
the signature theorem	$\operatorname{sign}(M) = \langle L_k,[M]\rangle$ for a closed manifold of dimension $4k$	Characteristic Classes
the $\hat A$-genus of a spin manifold	the characteristic number of the Dirac operator	Spin Geometry
the index of an elliptic operator	the difference of the dimensions of the kernel and cokernel	The Atiyah–Singer Index Theorem and K-Theory
the Atiyah–Singer index theorem	the index is the evaluation of a characteristic class of the symbol	The Atiyah–Singer Index Theorem and K-Theory
the index of the Dirac operator	$\hat A$-genus on a spin manifold, with the twisted version by a bundle	The Atiyah–Singer Index Theorem and K-Theory; Spin Geometry
the Riemann–Roch theorem	the holomorphic Euler characteristic as a Todd-class number	Algebraic Curves; The Atiyah–Singer Index Theorem and K-Theory
the cobordism invariance of the characteristic numbers	the numbers vanish on a boundary, defining the cobordism ring	Cobordism and Surgery Theory

The characteristic numbers are the topological invariants of a closed manifold, and they are the integrals that appear on the analytic side of the index theorems. The theorem of Atiyah and Singer computes the index of an elliptic operator from the symbol, and its special cases are the Gauss–Bonnet, signature and Riemann–Roch theorems, each of which identifies a characteristic number with an analytic index.

Classifying Spaces and K-Theory

The isomorphism classes of bundles over a base are classified by maps into a universal base, and K-theory is the cohomology theory in which the classes live.

Object	The property it has	Introduced in
the classifying space $BO(n)$, $BU(n)$	the base of the universal bundle, with $\mathbf{Vect}_n^{\mathbb{K}}(B) \cong [B, BO(n)]$	Characteristic Classes
the universal bundle and the universal classes	the bundle whose classes pull back to all the classes of the base	Characteristic Classes
the classifying map	the map $B \to BO(n)$ whose pullback is the bundle	Characteristic Classes
the Stiefel–Whitney and Chern classes as universal classes	the generators of $H^(BO(n))$ and $H^(BU(n))$	Characteristic Classes
the K-theory group $K^0(B)$	the Grothendieck group of the vector bundles, a cohomology theory	Topological K-Theory
the Chern character $K^0(B)\otimes\mathbb{Q} \cong H^{\mathrm{ev}}(B;\mathbb{Q})$	the rational isomorphism between K-theory and cohomology	Characteristic Classes; Topological K-Theory

The classifying space is the universal base of the rank-$n$ bundles, so that the characteristic classes are the pullbacks of the universal classes, and the homotopy classification of the bundles replaces the topological one. K-theory is the generalised cohomology theory in which the classes are the stable classes of bundles, and the index theorem is naturally a statement in it.

Warnings

An object that a reader may expect among the geometric bundles and invariants, and does not find, is recorded with the reason.

Object	Why it is not listed as a bundle or a characteristic class	Introduced in
a sheaf on a space	a presheaf with a gluing condition; a locally free sheaf is a bundle, but a general sheaf is not	Sheaves and the de Rham Complex
a distribution or a foliation	a subbundle of the tangent bundle with an integrability condition, not itself a bundle over the base	Fibre Bundles, Connections and Curvature
the Euler class of a non-orientable bundle	the construction needs an orientation; only the reduction modulo two survives	Characteristic Classes
the integral Chern classes of a flat complex bundle	they vanish in real cohomology by Chern–Weil, since the curvature is zero	Fibre Bundles, Connections and Curvature
a Möbius band as a trivial bundle	the non-orientable real line bundle over the circle, non-trivial	Topological K-Theory
the tangent bundle of $S^2$ as a trivial bundle	it is non-trivial, obstructed by the Euler class	Characteristic Classes

Summary

This article has listed the bundles of the corpus and the invariants that measure them. The fibre, vector and principal bundles open the list, with the transition functions, the associated bundles, the frame and tautological bundles and the line bundles; the connections and their curvature follow, with the connection form, the gauge transformation rule, the Bianchi identity, the holonomy and the flat case; the characteristic classes are recorded, Chern, Stiefel–Whitney, Pontryagin and Euler, with the Chern–Weil homomorphism, the splitting principle, the Chern character and the multiplicative sequences; the characteristic numbers, the Gauss–Bonnet, signature and Riemann–Roch theorems and the Atiyah–Singer index theorem follow; and the classifying spaces and K-theory close the list. Beside the examples stand the non-examples: a sheaf and a foliation are not bundles, a non-orientable bundle has no Euler class, and the real Chern classes of a flat bundle vanish.

Summary of Notation

A catalogue denotes its objects by name rather than by symbol. The symbols that appear in the tables are the following, and they are the symbols of the articles that introduce them.

Symbol	Meaning
$\pi : E \to B$, $g_{\alpha\beta}$, $P\times_GF$	Fibre bundle, transition functions, associated bundle
$TM$, $T^*M$, $\Gamma(E)$, $K_M$	Tangent and cotangent bundles, sections, canonical bundle
$\nabla$, $\omega$, $R$, $\Omega$	Connection, connection form, curvature, curvature form
$c_i$, $w_i$, $p_i$, $e(E)$	Chern, Stiefel–Whitney, Pontryagin and Euler classes
$\operatorname{ch}$, $L_k$, $\hat A$, Todd	Chern character, $L$-polynomial, $\hat A$-genus, Todd class
$BO(n)$, $BU(n)$, $K^0(B)$	Classifying spaces and K-theory
$\langle\kappa(TM),[M]\rangle$	Characteristic number
$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{C}$	The standard number systems of the corpus
$\operatorname{Ad}$, $\operatorname{End}$	Adjoint action of the structure group on $\mathrm{G}$; endomorphism bundle of a vector bundle
$\mathbb{K}$	The field of scalars of a bundle or of a classifying space
$\mathcal{F}$, $\operatorname{sign}(M)$	The curvature of a line bundle, $\frac{i}{2\pi}\mathcal{F} = c_1(L)$; the signature of a closed manifold of dimension $4k$