# Studying characteristic classes *Teaching written and self-checked by GPT-6.1 Sol (OpenAI), at Ultra. Independently authored guide dedicated under CC0. Individual lessons identify their human sources and any licensed adaptations. Independent review remains separate.* The course develops several ways to extract information from a vector bundle or a manifold. A useful reading path follows the question being asked: constructing a bundle, ruling out a geometric structure, evaluating an integral class, computing a form from curvature, or comparing manifolds by bordism. The complete proofs and solved exercises remain in the linked chapters. The identifiers in those links are stable addresses; the order below is a reading order. ## Build a bundle and give its classes a meaning Begin with [Vector bundles and their constructions](DG-CHAR-01.html), then [Grassmannians and classifying maps](DG-CHAR-03.html). These chapters explain local triviality, metrics, pullbacks, complements and classification. In the [Thom and Euler chapter](DG-CHAR-06.html), a local fibre orientation becomes a global relative cohomology class. Its chain, product, excision and coefficient proofs also supply the homological tools used later. The [Gysin and projective-splitting chapter](DG-CHAR-08.html) calculates the projective-fibre cohomology and proves the injectivity of flag pullback. Then [Steenrod squares and Stiefel–Whitney classes](DG-CHAR-05.html) constructs the operations and classes. At this point a real characteristic-class computation has a proved definition and product rule. ## Use a class to test a geometric construction Continue immediately with [Projective tangent bundles and their obstructions](DG-CHAR-02.html). Its tangent bundle, inverse class and characteristic-number calculations address three different questions: an actual frame, the necessary normal rank of an immersion, and the possibility of a boundary. The projective rank-budget examples explain how to choose the test and how much its vanishing establishes. Fundamental classes and the boundary sign are constructed before they are used. Return to [Schubert cells and universal Grassmannian cohomology](DG-CHAR-04.html) to determine the universal rings. Its symmetric-polynomial proof will later identify integral and rational generators, beyond the projective test cases. The [Chern-class chapter](DG-CHAR-09.html) then passes from line normalization to Whitney, conjugation, tensoring, integral evaluation and the universal complex ring. Its Hopf-circle calibration explains why an arbitrary simplifying pullback cannot replace an injective flag pullback. The [Pontryagin-class chapter](DG-CHAR-10.html) records the exact integral and two-torsion qualifications for real bundles. [Manifold duality, the diagonal and Wu classes](DG-CHAR-07.html) explains the evaluation pairings. The [frame-obstruction companion](DG-CHAR-08B.html) then gives the geometric section and Euler obstructions, with the hypotheses under which their vanishing is sufficient. ## Compare integral topology with differential forms Read [Connections, curvature and characteristic forms](DG-CHAR-17.html) here, once the integral classes and duality are available. It proves de Rham comparison, transgression, Chern–Weil, the Euler–Pfaffian identification and generalized Gauss–Bonnet. Curvature gives the real image of an integral class. The earlier integral construction remains necessary to retain torsion and normalization. For a complex bundle, compare a class calculated through line splitting with the form calculated from a connection. For an oriented tangent bundle, compare its Euler evaluation with the Pfaffian integral. These comparisons use the proved compatibility theorems in the curvature chapter; a form calculation alone does not recover the full integral class. The course's detailed characteristic-form proofs remain separate from the principal-connection and holonomy foundations identified in that chapter. ## Turn evaluations into a comparison of manifolds [Characteristic numbers and projective-product independence](DG-CHAR-11.html) organizes evaluations by partitions and proves the projective independence matrices. [The oriented cobordism ring](DG-CHAR-12.html) proves the boundary and gluing facts. [Thom spaces and the Pontryagin–Thom construction](DG-CHAR-13.html) then turns geometric bordism into a homotopy problem. The three companions [Homotopy fibres and the Serre spectral sequence](DG-CHAR-13B.html), [Rational homotopy and the Hurewicz range](DG-CHAR-13C.html), and [Rational oriented bordism and projective generators](DG-CHAR-13D.html) supply the actual comparison proofs and the complete rational polynomial ring. They are substantive chapters, with their own exercises, rather than prerequisites left to an external book. The separate [Stiefel–Whitney numbers and unoriented bordism](DG-CHAR-13E.html) proves detection in every dimension. It supplies the complete mod-two operation basis, one-group calculation and Thom-module argument required for that converse. Rational oriented detection and mod-two unoriented detection are different comparison theorems; retain their coefficient and orientation conditions when applying them. ## Recover signature and compare geometric structures [Multiplicative sequences and the signature theorem](DG-CHAR-14.html) compares an intersection matrix with Pontryagin numbers. Its two-example calibration explains why independent projective products determine the degree-eight polynomial. The complete form and boundary arguments, polynomial construction and formal projective coefficient calculation prove the signature theorem in all dimensions. The [odd-prime reduced-power companion](DG-CHAR-14B.html) supplies the precise Wu polynomial and its homotopy conclusions, including the stated individual modulo-three case. [Combinatorial L-classes and piecewise linear fibres](DG-CHAR-15.html) constructs rational classes from fibre signatures. [Smooth fibres, triangulation comparison and lens spaces](DG-CHAR-15B.html) compares this construction with smooth characteristic classes and computes torsion examples. Milnor's exotic seven-spheres provides the complete smooth-structure application. Finally [Smooth triangulations and the integral PL obstruction](DG-CHAR-15C.html) proves the needed triangulation and extension arguments and distinguishes rational invariance from a universal integral refinement. The full reading path contains all twenty-five chapters and all 155 existing solved exercises. For a particular application, follow the exact proved dependencies in its chapter rather than assuming that adjacent chapters use every earlier result. Human mathematical sources are credited at the relevant constructions; marked Roberts adaptations retain CC BY 4.0 and the accompanying licence.