Topology, Bundles, and Global Geometric Tools

Local differential geometry cannot answer every geometric question. A manifold may be covered by ordinary coordinate balls yet contain noncontractible loops; a bundle may look like a product in every chart yet be globally twisted; a field may exist locally but be obstructed globally. This chapter supplies a bounded toolkit for recognizing those situations: elementary homotopy and homology, transition cocycles, principal and associated bundles, characteristic classes, spin lifts, and Morse functions.

Helpful background. Calculus and Linear Algebra supplies quotient spaces, matrix groups, determinants, and quadratic forms; Tensor Fields and Natural Operations supplies smooth tensor fields, pullbacks, and the tangent- and cotangent-bundle examples; Exterior Algebra and Differential Forms supplies forms, wedge products, and pullbacks; The Riemann Curvature Tensor supplies the familiar tangent-bundle curvature model generalized later in the chapter.

From local charts to global obstructions

The recurring pattern is gluing. Local pieces are simple, but their overlap maps must satisfy cocycle conditions. Changing local trivializations changes the cocycle representative without changing the global object. Some cocycles can be removed by such changes, while others define nontrivial topology. Homotopy and homology measure global cycles; characteristic classes turn bundle gluing into cohomological obstructions; Morse theory reconstructs topology from changes in sublevel sets.

Three distinctions organize the chapter. First, a local potential is not a global connection, and a global curvature form need not have a global potential. Second, vanishing of one obstruction does not remove later ones: orientability, time orientation, and spin-lift existence are separate. Third, a differential-form representative sees real cohomology but can miss torsion. The local equation $dF=0$ therefore cannot by itself classify every bundle or flux sector.

Hatcher develops homotopy, homology, and cohomology with the sphere and torus among the central examples in Hatcher 2002, Chapters 1–3, pp. 21–260. The vector-bundle and characteristic-class interface is developed separately in Hatcher 2017, §§1.1–1.2 and 3.1–3.3, pp. 6–37 and 77–106.

Preparation and repair routes

Readers need smooth manifolds, maps, tangent bundles, differential forms, and Stokes’ theorem. If cycles and exact forms are unfamiliar, begin with de Rham Cohomology and Global Flux before entering this chapter. Linear algebra at the level of bases, matrix groups, and quotient spaces is enough for the bundle pages.

Curvature is helpful but not required for the opening pages. A reader who knows the Riemann tensor but not bundles should first learn transition functions and principal bundles; the connection page then reveals the Riemann tensor as one instance of a broader bundle-curvature construction. Spinors are not assumed. The obstruction page defines only the lift needed for their global existence, not the representation theory or Dirac equation.

Route through topology and vector bundles

The first route asks what information survives every change of coordinates:

1. Homotopy, Homology, and Geometric Invariants computes $\pi_1$, low-degree homology, cohomology periods, and Euler characteristic for $S^1$, $S^2$, and $T^2$.
2. Vector Bundles and Transition Functions builds bundles from local products and cocycles, then distinguishes the cylinder from the Möbius line bundle.
3. Principal and Associated Bundles makes local frame freedom global and reconstructs vector bundles from representations of the structure group.

This route is enough to explain why a zero section does not trivialize a vector bundle, why a global frame does, and why a principal bundle has a global section precisely when it is trivial. Its recurring test is consistency around a closed loop: the local representatives must satisfy the final overlap identification even when every intermediate neighborhood is trivial. Parallel transport around that loop requires the additional choice of a connection.

The frame bundle is the bridge back to geometry. Tensor bundles are associated to representations of $GL(n,\mathbb R)$; orthonormal frames reduce the group using a metric; orientation, spin, and connections add distinct layers of structure. None is supplied merely by writing local components.

Route through connections and obstructions

The second route asks what local geometric data can reveal about global gluing:

1. Bundle Connections and Characteristic Classes defines horizontal transport, derives the curvature transformation law, and computes integer $U(1)$ flux on $S^2$.
2. Orientability, Spin Structures, and Global Obstructions separates $w_1$, Lorentzian time orientation, $w_2$, and the multiplicity of spin structures.
3. Morse Theory and Global Geometric Constructions relates nondegenerate critical points to handle attachment and checks Euler characteristic on the sphere and torus.

Gaiter states the Chern–Weil closedness and connection-independence results in Gaiter 2024, §§5.2–5.3, pp. 27–36. Wernli constructs the spin-lift cocycle and its $w_2$ obstruction in Wernli 2019, §2.3.1, Theorem 2.3.15, pp. 60–66. Milnor develops the Morse normal form and topology change across critical levels in Milnor 1963, §§1–5, pp. 1–31.

These results have different hypotheses. Chern–Weil forms represent real cohomology classes and miss torsion. A spin lift presupposes the relevant oriented orthonormal frame bundle. A Morse handle statement needs an isolated nondegenerate critical value and compactness or properness sufficient to control the flow.

Three recurring calculations

The simplest global invariant calculation compares

$$\pi_1(S^1)=\mathbb Z, \qquad \pi_1(S^2)=0, \qquad \pi_1(T^2)=\mathbb Z^2.$$

Homology adds the two-cycle of $S^2$ and the independent one-cycles of $T^2$. Their Euler characteristics are $0$, $2$, and $0$, respectively. These values reappear in Morse critical-point counts, providing an independent check.

The bundle calculation glues an interval’s endpoint fibers with $+1$ or $-1$. The first choice gives the cylinder; the second gives the Möbius line bundle. Both are locally trivial, but the Möbius bundle has no nowhere-zero global section. This is the elementary model for a transition cocycle that cannot be removed globally.

The curvature calculation covers $S^2$ with two patches whose $U(1)$ potentials differ by $n\,d\varphi$. Their common curvature has normalized flux

$$\frac{1}{2\pi}\int_{S^2}F=n\in\mathbb Z.$$

The integer agrees with the winding of the overlap function. It also proves that nonzero-flux curvature cannot be $dA$ for one global potential, since Stokes’ theorem would then force zero flux.

Boundaries of the toolkit

These pages do not classify manifolds, bundles, or Lorentzian spacetimes in general. Equal homology does not imply homeomorphism. Characteristic classes do not contain every bundle invariant. A spin structure permits a spinor bundle but does not select a spinor field. A Morse function is not automatically a Lorentzian time function, and its critical points do not by themselves imply causal or geodesic singularities.

The chapter also does not develop global causal theorems, index theory, anomalies, or gauge-field dynamics. Its purpose is diagnostic: identify which proposed global construction needs a cocycle, which obstruction class must vanish, which integral is quantized, and which compactness hypothesis controls a deformation.

Chapter diagnostic

A real line bundle over $S^1$ is formed by $(0,v)\sim(2\pi,-v)$. Determine whether it has a nowhere-zero section, compute the relevant first cohomology group that permits two $\mathbb Z_2$ gluing choices, and explain why no local chart can distinguish the twisted bundle from the product bundle.

Solution

A section is represented by a continuous function $f:[0,2\pi]\to\mathbb R$ satisfying $f(2\pi)=-f(0)$. If $f(0)\ne0$, the endpoint values have opposite signs, so the intermediate value theorem forces a zero. Thus no nowhere-zero section exists and the line bundle is nontrivial.

The group $H^1(S^1;\mathbb Z_2)\cong\mathbb Z_2$ contains the trivial and sign-twisted classes. On any contractible arc, one can choose a local frame and identify the bundle with the product. Only after following the transition around the full circle does the sign survive. This simultaneously checks the cocycle, section, and local-versus-global viewpoints.

What this chapter enables

After the topology route, a reader can compare basic manifolds using $\pi_1$, homology, periods, and Euler characteristic; construct a vector bundle from transition functions; and translate between principal bundles, frames, and associated fields. After the obstruction route, a reader can derive local gauge transformations of a connection, verify curvature covariance, compute an integral characteristic number, state the $w_1$ and $w_2$ tests, count spin structures using $H^1(M;\mathbb Z_2)$, and recognize the compactness assumptions behind a Morse handle argument.

References

• Gaiter, Jacob. Characteristic Classes, Principal Bundles, and Curvature. University of Texas at Austin notes, 2024. §§5.1–5.3, pp. 22–36.
• Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2002. Chapters 0–3, pp. 1–260.
• Hatcher, Allen. Vector Bundles and K-Theory. Version 2.2, 2017. §§1.1–1.2 and 3.1–3.3, pp. 6–37 and 77–106.
• Milnor, John. Morse Theory. Annals of Mathematics Studies 51, Princeton University Press, 1963. Open PDF, §§1–6, pp. 1–38.
• Wernli, Konstantin. Lecture Notes on Spin Geometry. arXiv:1911.09766, 2019. §2.3.1, pp. 60–67.