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For the vector-bundle curvature matrix $R$ of a metric connection on a vector bundle on a real oriented vector bundle, the A-hat form is the even differential form defined by the displayed finite-degree power series, with square root having constant term one. The expression is independent of the choice of orthonormal frame. Equivalently, for formal paired curvature Chern roots $\pm x_j$, it is ${\prod }_j(x_j/2)/\sinh (x_j/2)$. The Bianchi identity makes it closed, and changing the connection changes it by an exact differential form via Chern-Weil connection transgression. Its degree-four part is $-p_1/24$, where $p_1=-\mathrm{tr}(R\wedge R)/(8{\pi }^2)$ is the first Pontryagin class form.

The Chern-Weil homomorphism turns a polynomial in curvature into a topological cohomology class. For a principal bundle with Lie group $G$, a degree-$k$ polynomial $P$ on its Lie algebra that is invariant under the adjoint action determineswhere $F$ is the curvature form of any principal connection. The Bianchi identity makes $P(F)$ a closed differential form. Changing the principal connection changes it by an exact differential form, so the de Rham cohomology class depends on the principal bundle, not that choice. Products of polynomials become cup products.^[ref]

For example, the trace-square polynomial produces $\mathrm{tr}(F\wedge F)$, underlying the first Pontryagin class. This is the bridge between the locally measured curvature and a global topological invariant.

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]

James Simons worked on differential geometry and geometric invariants. With Shiing-Shen Chern, he introduced the secondary characteristic forms now associated with Chern-Simons forms.^[ref]

The construction produces a form whose exterior derivative is a characteristic curvature form. The introduction explains how a search for a combinatorial formula for a number obtained from a Pontryagin class led to an interesting boundary term. That term became a geometric object in its own right, illustrating how an obstruction in one calculation can generate a new invariant.^[ref]

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.

The stabilizer of a line in $O(n)$ is $O(1)\times O(n-1)$, giving the homogeneous space ${\mathbb{RP}}^{n-1}$. Its classifying-space Serre fibration has rationally acyclic fiber for odd $n$, so its cohomological Serre spectral sequence has only its degree-zero row and gives the displayed restriction isomorphism. The factor $BO(1)={\mathbb{RP}}^{\infty }$ is itself rationally acyclic. For even $n$, the fiber has a top rational class, with determinant-sign monodromy, so the one-row argument fails. The top universal Pontryagin class $p_{n/2}$ is nonzero on $BO(n)$ but vanishes on restriction to $BO(n-1)$, witnessing genuine failure of stabilization.