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For a complex vector bundle $E\to X$ of rank $r$, its Chern classes are integral cohomology classesThey are natural under pullback and obey the Whitney sum formula for Chern classes. For a complex line bundle, $c_1$ is the Euler class of its underlying oriented real two-plane bundle; generally $c_r(E)$ is the Euler class of the underlying real bundle.^[ref]

For example, if $h=c_1(\mathcal{O}(1))$ on Complex projective space, the Euler sequence gives $c(T{\mathbb{P}}^n)=(1+h{)}^{n+1}$, truncated above degree $2n$. For an algebraic vector bundle on a smooth variety, algebraic Chern classes instead belong to the Chow groups; the cycle class map recovers the topological classes over the complex numbers.

James D. Stasheff works on topology and its algebraic structures. With John Milnor, he wrote Characteristic Classes by John Milnor and James D. Stasheff, developing invariants that detect how vector bundles twist over a space.^[ref]

The total Pontryagin class illustrates the importance of coefficient conventions: its product formula is stated modulo elements of order two in their theorem. Keeping that qualification prevents an integral statement from being inferred from a result valid only after removing a torsion ambiguity.^[ref]

John Milnor works across topology, geometry and dynamical systems. His expositions connect local calculations with global structure: Morse Theory by John Milnor develops the topology of a manifold through critical points, while his complex-dynamics lectures study iteration of holomorphic maps.^[ref][ref]

Near a nondegenerate critical point, the Morse lemma reduces a smooth function exactly to a signed quadratic form in suitable coordinates. Its index controls the topological change in sublevel sets. With James D. Stasheff, Milnor also wrote Characteristic Classes by John Milnor and James D. Stasheff, developing invariants of vector bundles.

For a real vector bundle $E$, its Pontryagin classes areThey use the even-index Chern classes of the complexification. An integral qualification matters for direct sums: for the total Pontryagin class,Thus the usual multiplicative formula holds over the rational numbers, over the real numbers, or after discarding elements of order two. It need not hold without qualification in integral cohomology. The missing terms are products of odd Chern classes of complexified real vector bundles, which are annihilated by two.^[ref]

For a metric connection on a vector bundle with curvature matrix $R$, a real representative isThe Bianchi identity makes this a closed differential form, and Chern-Weil connection transgression makes its de Rham cohomology class independent of the connection on a vector bundle. This normalization gives the degree-four A-hat form $-p_1/24$. Curvature representatives record real classes and therefore cannot detect the integral torsion qualification.

Let $S$ be a compact, oriented, boundaryless embedded submanifold of dimension $n$ in an oriented smooth manifold $M$ of dimension $2n$. Its self-intersection number is the signed intersection number of $S$ with a small transverse push-off. A transverse section of a fiber bundle of the normal bundle gives that push-off; its signed zeros show thatThe Euler class makes the answer independent of the chosen transverse section of a fiber bundle.^[ref]

A line in the Complex projective plane has square $1$. A fiber ${\mathbb{P}}^1\times \{p\}$ in ${\mathbb{P}}^1\times {\mathbb{P}}^1$ has square zero because it has a disjoint homologous push-off. The exceptional divisor of a blowup of a smooth algebraic surface has square $-1$. For noncompact embedded submanifolds or embedded submanifolds with boundary, an integer requires additional support or boundary data; the unrestricted signed-count definition does not apply.