Fact-checked by Grok 2 weeks ago

Pontryagin class

In algebraic topology, the Pontryagin classes are a sequence of characteristic classes p_k \in H^{4k}(X; \mathbb{Z}) associated to real vector bundles over a space X, providing invariants that capture topological information about the bundle's structure.^[1] Named after the Soviet mathematician Lev Pontryagin, who developed them as an analogue to Chern classes for real bundles, these classes are fundamental tools for classifying vector bundles and studying the topology of manifolds.^[1] They are defined for bundles of any rank and play a key role in distinguishing bundles that are stably equivalent but not isomorphic. The Pontryagin classes are most naturally defined using the complexification of a real vector bundle \xi: specifically, p_k(\xi) = (-1)^k c_{2k}(\xi \otimes \mathbb{C}), where c_{2k} denotes the $2k-th Chern class of the complexified bundle \xi \otimes \mathbb{C}, and the classes live in cohomology with integer coefficients (or rational coefficients for computational purposes).^[1]^[2] This construction ensures that the classes are integral and stable under direct sums, with p_k(\xi) = 0 for k > \dim(\xi)/2.^[2] Under reduction modulo 2, the Pontryagin classes relate to Stiefel-Whitney classes via p_k(\xi) \equiv w_{2k}(\xi)^2 \pmod{2}, linking them to mod-2 invariants.^[1] For oriented bundles of even rank $2m, the top Pontryagin class satisfies p_m(\xi) = e(\xi)^2, where e(\xi) is the Euler class.^[1]^[2] A defining property of Pontryagin classes is their multiplicativity under Whitney sums: the total Pontryagin class p(\xi \oplus \eta) = p(\xi) \cdot p(\eta), where the product is in the cohomology ring, allowing recursive computation for sums of bundles.^[1]^[2] They are natural under pullbacks, meaning f^* p_k(\xi) = p_k(f^* \xi) for a continuous map f, and generate the cohomology ring of the classifying space BSO(n) as a polynomial algebra on p_1, \dots, p_{\lfloor n/2 \rfloor} (with the Euler class adjoined for even n).^[2] These classes exhibit divisibility properties in manifold contexts, such as the first Pontryagin number of a smooth oriented 4-manifold being divisible by 3.^[2] Pontryagin classes have profound applications in differential topology and cobordism theory, where the Pontryagin numbers—integrals of powers of these classes over a manifold—serve as complete invariants for oriented manifolds in certain dimensions, as established by Thom and Milnor.^[2] For instance, they feature in Hirzebruch's signature theorem, relating the signature of a 4k-manifold to combinations of Pontryagin classes via L-genus polynomials, such as L_1 = p_1/3 and L_2 = (7p_2 - p_1^2)/45.^[2] In Milnor's 1956 analysis of exotic 7-spheres, Pontryagin classes helped prove the existence of manifolds homeomorphic but not diffeomorphic to the standard sphere.^[2] More broadly, they underpin obstructions to bundle existence and manifold embeddings, connecting to K-theory via the Chern character and influencing modern areas like equivariant topology.^[1]

Definition and Axioms

Formal Definition

The Pontryagin classes of a real vector bundle E of rank n over a topological space X are cohomology classes p_k(E) \in H^{4k}(X; \mathbb{Z}) for k \geq 1, with the zeroth class defined by convention as p_0(E) = 1. The total Pontryagin class is the formal sum p(E) = 1 + p_1(E) + p_2(E) + \cdots + p_m(E), where m = \lfloor n/2 \rfloor, and higher classes vanish, i.e., p_k(E) = 0 for k > n/2. This structure ensures that the Pontryagin classes capture topological invariants of the bundle in even-degree cohomology with integer coefficients.^[3] These classes are defined topologically via the classifying space BO(n) for the orthogonal group O(n), which classifies stable real vector bundles of rank n. Given a classifying map f: X \to BO(n) corresponding to the bundle E, the Pontryagin classes are the pullbacks p_k(E) = f^*(\tilde{p}_k), where \tilde{p}_k \in H^{4k}(BO(n); \mathbb{Z}) are the universal Pontryagin classes. The rational cohomology ring H^*(BO(n); \mathbb{Q}) is a polynomial algebra over \mathbb{Q} generated by these universal classes up to the appropriate degree, while the integral cohomology H^*(BO(n); \mathbb{Z}) includes 2-torsion and is generated by the Pontryagin classes together with the Stiefel-Whitney classes subject to relations.^[3]^[4] Lev Pontryagin introduced these classes in the 1940s as characteristic cycles on differentiable manifolds, initially motivated by problems in homotopy theory where they serve as obstructions to certain extensions or sections of bundles.^[5]^[3]

Axiomatic Properties

Pontryagin classes for real vector bundles are uniquely characterized as cohomology classes p_k \in H^{4k}(-; \mathbb{Z}) by three fundamental axioms: naturality, multiplicativity, and normalization.^[6] Naturality requires that Pontryagin classes commute with pullbacks along continuous maps between base spaces. Specifically, for a map f: X \to Y and a real vector bundle \xi over Y, the classes satisfy p_k(f^*\xi) = f^* p_k(\xi) for each k \geq 1. This ensures that the classes are functorial and depend only on the isomorphism class of the bundle.^[6] Multiplicativity is captured by the Whitney sum formula for the total Pontryagin class p(\xi) = 1 + p_1(\xi) + p_2(\xi) + \cdots, which states that p(\xi \oplus \eta) = p(\xi) \cup p(\eta) for any real vector bundles \xi and \eta over the same base. In components, this implies p_k(\xi \oplus \eta) = \sum_{i+j=k} p_i(\xi) \cup p_j(\eta). This property reflects the ring structure in the cohomology of classifying spaces and holds modulo 2 even without orientation assumptions.^[6] Normalization specifies that the total class of the trivial bundle is 1, so p_k(\epsilon^n) = 0 for all k > 0 and any trivial bundle \epsilon^n, while p_0 = 1. Additionally, the classes vanish above half the bundle rank, p_k(\xi) = 0 for k > \operatorname{rank}(\xi)/2, and they are defined such that the universal Pontryagin classes over the Grassmannian generate the appropriate cohomology ring.^[6] These axioms determine the Pontryagin classes uniquely up to isomorphism as natural transformations from the functor of real vector bundles to cohomology groups H^{4k}(-; \mathbb{Z}). The uniqueness follows from the theory of multiplicative cohomology operations, where the axioms fix the classes as the unique sequence satisfying the product formula and normalization on universal bundles over BO(n). This is established via the splitting principle, which reduces computations to formal sums of line bundles, allowing the classes to be expressed in terms of formal power series associated to the bundle's structure.^[6] In the broader context of characteristic classes, Pontryagin classes are distinguished from Chern classes by their application to real (rather than complex) bundles and by their degrees, which are multiples of 4 rather than 2, while both use integer coefficients. Unlike Stiefel-Whitney classes, which take values in mod-2 cohomology across all degrees and satisfy similar axioms but with \mathbb{Z}/2\mathbb{Z} coefficients, Pontryagin classes capture integral invariants relevant to oriented structures and cobordism theory.^[6]

Constructions

Via Classifying Spaces

The Pontryagin classes of a real vector bundle can be constructed using the topology of classifying spaces, specifically through the infinite Grassmannian manifolds that classify stable real vector bundles. The infinite Grassmannian \mathrm{Gr}_n(\mathbb{R}^\infty) parametrizes n-dimensional subspaces of \mathbb{R}^\infty, and over this space lies the universal (tautological) bundle \gamma^n, whose total space consists of pairs (V, v) where V \subset \mathbb{R}^\infty is an n-plane and v \in V. The cohomology ring H^*(\mathrm{Gr}_n(\mathbb{R}^\infty); \mathbb{Z}) is generated by the Pontryagin classes p_1(\gamma^n), \dots, p_{\lfloor n/2 \rfloor}(\gamma^n) of this tautological bundle, along with relations arising from the geometry of the Grassmannian. These classes reside in degrees $4k and provide the universal representatives for Pontryagin classes of all real vector bundles of rank n.^[3] For a smooth manifold X and a real vector bundle E \to X of rank n, the bundle E is classified by a map f: X \to \mathrm{Gr}_n(\mathbb{R}^\infty) (up to homotopy), which pulls back the universal bundle via f^* \gamma^n \cong E. The Pontryagin classes of E are then defined as the pullbacks p_k(E) = f^* p_k(\gamma^n) for k = 1, \dots, \lfloor n/2 \rfloor, lying in H^{4k}(X; \mathbb{Z}). This construction ensures that the classes are stable under Whitney sums in the sense that adding trivial line bundles does not alter them, reflecting the homotopy equivalence between \mathrm{Gr}_n(\mathbb{R}^\infty) and the classifying space \mathrm{BO}(n).^[3] In the stable regime, where the rank n is sufficiently large compared to the dimension of X, the Pontryagin classes stabilize, meaning p_k(E) for k \leq m depends only on the stable class of E in the reduced K-theory \tilde{KO}(X). The direct limit \mathrm{BO} = \varinjlim_{n \to \infty} \mathrm{BO}(n) = \varinjlim_{n \to \infty} \mathrm{Gr}_n(\mathbb{R}^\infty) serves as the classifying space for stable real vector bundles, and its integral cohomology ring is the polynomial algebra H^*(\mathrm{BO}; \mathbb{Z}) \cong \mathbb{Z}[p_1, p_2, \dots ], where each |p_k| = 4k. This ring structure underscores the generators' role in capturing the Pontryagin characteristic classes for real bundles in the stable range.^[3]

From Chern Classes

One method to compute the Pontryagin classes of a real vector bundle E over a space B involves complexification, yielding the complex vector bundle E \otimes \mathbb{C}, which decomposes as E^{1,0} \oplus E^{0,1} where E^{0,1} = \overline{E^{1,0}} in the presence of a compatible almost complex structure, though the relation holds generally. The Chern classes c_k(E \otimes \mathbb{C}) of this complexified bundle determine the Pontryagin classes p_k(E) via the formula

p_k(E) = (-1)^k c_{2k}(E \otimes \mathbb{C})

modulo 2-torsion in the cohomology ring H^{4k}(B; \mathbb{Z}).^[7] This relation arises because the formal Chern roots \alpha_1, \dots, \alpha_m of E \otimes \mathbb{C} (with m = \dim_{\mathbb{R}} E) satisfy that the odd-degree symmetric polynomials vanish modulo 2-torsion, while the even-degree ones encode the Pontryagin classes through symmetric functions of the squares \alpha_j^2.^[3] For explicit computation, the relation can be expanded using Newton-Girard identities to express p_k(E) as polynomials in the Chern classes c_i(E \otimes \mathbb{C}). In low degrees, this yields p_1(E) = -c_2(E \otimes \mathbb{C}), and more generally for the underlying real bundle of a complex vector bundle V, the formula adjusts to account for the decomposition V_R \otimes \mathbb{C} \cong V \oplus \overline{V}, giving

p_1(V_R) = c_1(V)^2 - 2 c_2(V).

Higher-degree classes follow similarly; for instance, p_2(V_R) = c_2(V)^2 - 2 c_1(V) c_3(V) + 2 c_4(V), derived from the product of Chern classes c(V) \cdot c(\overline{V}).^[8] These polynomial expressions facilitate algebraic manipulation without direct recourse to classifying spaces.^[3] This approach is particularly advantageous for computations on complex manifolds, where the Chern classes of the holomorphic tangent bundle T^{1,0}M are often primary invariants (e.g., via the Atiyah-Singer index theorem or Todd genus), allowing Pontryagin classes of the real tangent bundle TM to be derived efficiently from them. For example, on \mathbb{CP}^2, the Chern classes yield c(T\mathbb{CP}^2) = (1 + x)^3 with x the generator of H^2(\mathbb{CP}^2; \mathbb{Z}), leading to p_1(TM) = 3x^2.^[7] The modulo 2-torsion caveat ensures integrality, as Pontryagin classes live in integral cohomology, but the relation holds integrally up to this adjustment in many geometric contexts.^[3]

Using Gauge Curvature

In the differential-geometric construction of Pontryagin classes, a real vector bundle E over a smooth manifold M is equipped with a connection \nabla, which induces a curvature 2-form \Omega \in \Omega^2(M, \mathfrak{[gl](/page/GL)}(r, \mathbb{R})), where r = \rank(E). The Pontryagin characteristic forms are the images under the Chern-Weil homomorphism of the invariant polynomials on the Lie algebra \mathfrak{[gl](/page/GL)}(r, \mathbb{R}) that correspond to the Pontryagin classes. For example, the first Pontryagin form is the closed 4-form

P_1(\nabla) = -\frac{1}{8\pi^2} \Tr(\Omega \wedge \Omega),

and higher P_k(\nabla) are polynomials in traces of powers of \Omega.^[9] This construction arises from applying Chern-Weil theory to the universal invariant polynomials defining the Pontryagin classes.^[10] The closedness of P_k(\nabla) follows from the Bianchi identity d_\nabla \Omega = 0, which implies d P_k(\nabla) = 0. By the de Rham theorem, which identifies de Rham cohomology with singular cohomology with real coefficients, the cohomology class [P_k(\nabla)] \in H^{4k}_{dR}(M; \mathbb{R}) represents the Pontryagin class p_k(E) \in H^{4k}(M; \mathbb{R}). Chern-Weil theory further ensures that this class lies in the image of integral cohomology H^{4k}(M; \mathbb{Z}), providing the integrality of Pontryagin classes.^[10]^[11] The form P_k(\nabla) is invariant under gauge transformations, as the trace of the adjoint action on \Omega preserves the polynomial invariants. Moreover, it is independent of the choice of connection \nabla: for two connections \nabla_1 and \nabla_2 with curvatures \Omega_1 and \Omega_2, the difference P_k(\nabla_1) - P_k(\nabla_2) is exact, d \alpha_k(\nabla_1, \nabla_2) for some transgression form \alpha_k.^[9]^[10] For non-flat connections where \Omega \neq 0, the Pontryagin forms yield non-trivial primary classes, but the transgression forms \alpha_k relate to secondary characteristic classes, which refine the invariants by incorporating connection data when primary classes vanish (e.g., for flat bundles). These secondary classes, introduced in the context of differential characters, capture the differential-geometric structure beyond topology.^[10]

Core Properties

Naturality and Multiplicativity

The Pontryagin classes are natural under pullbacks. Specifically, for a continuous map \psi: X \to Y, \psi^* p_k(\eta) = p_k(\psi^* \eta) for each k, where \eta is a real vector bundle over Y and p_k denotes the k-th Pontryagin class in H^{4k}(Y; \mathbb{Z}). This property follows from the axiomatic definition of Pontryagin classes as natural transformations from the functor of real vector bundles to cohomology, ensuring compatibility with pullbacks and making them well-defined invariants under continuous maps.^[12] A key feature of Pontryagin classes is their multiplicativity under the Whitney sum operation. For real vector bundles E and F over the same base, the total Pontryagin class satisfies p(E \oplus F) = p(E) \cup p(F) in the cohomology ring, where the cup product corresponds to the ring structure. This Whitney sum formula extends the multiplicativity axiom, implying that the individual classes are determined by \sum_{i+j=k} p_i(E) \cup p_j(F) = p_k(E \oplus F). The formula holds integrally, reflecting the integral nature of the classes.^[8] For direct sums, the multiplicativity directly yields the Whitney sum rule above. The tensor product rule is more involved: the Pontryagin classes p_k(E \otimes F) can be expressed in terms of the Pontryagin classes p(E), p(F), and the Stiefel-Whitney classes w(E), w(F) via the complexification and splitting principle, accounting for the real structure and torsion elements in the odd Chern classes. Explicitly, since p_k(E \otimes F) = (-1)^k c_{2k}((E \otimes F) \otimes \mathbb{C}) and the Chern classes of the tensor product of complexifications are symmetric functions of paired roots, the Stiefel-Whitney classes enter to resolve the 2-torsion contributions from the real bundle constraints. These properties have significant consequences for pullbacks and stable equivalence of bundles. Naturality ensures that Pontryagin classes are preserved under pullback along any map, facilitating computations on classifying spaces and fiber bundles. For stably equivalent bundles, where E \oplus \epsilon^m \cong F \oplus \epsilon^n for trivial bundles \epsilon^k, the multiplicativity implies p(E) = p(F) since p(\epsilon^k) = 1, making Pontryagin classes stable invariants that classify bundles up to stable isomorphism.

Pontryagin Classes of Manifolds

For a smooth n-manifold M, the Pontryagin classes p_k(TM) of its tangent bundle TM lie in the cohomology group H^{4k}(M; \mathbb{Z}) for each k \geq 1, with the total Pontryagin class given by p(TM) = 1 + p_1(TM) + p_2(TM) + \cdots.^[6] These classes provide topological invariants that encode information about the bundle's structure, extending the axiomatic definition to the geometric setting of the manifold's tangent space. The total Pontryagin class p(TM) is instrumental in determining the minimal embedding dimension of M into Euclidean space, as analyzed in Hirsch-Smale theory, where the theory equates smooth embeddings up to regular homotopy with monomorphisms in the stable homotopy category of bundles, and Pontryagin classes serve as primary obstructions to such embeddings. For instance, if the stable normal bundle's classes fail to invert p(TM) appropriately in cohomology, higher codimensions are required for embeddability.^[6] In the context of embeddings i: M \hookrightarrow \mathbb{R}^{n+q}, the normal bundle \nu satisfies TM \oplus \nu \cong \varepsilon^{n+q}, the trivial bundle of rank n+q, implying p(TM \oplus \nu) = p(\varepsilon^{n+q}) = 1.^[6] By the multiplicativity of Pontryagin classes under Whitney sum, this yields p(TM) \cup p(\nu) = 1 in H^*(M; \mathbb{Z}), constraining the possible normal bundles and thus the embedding dimensions.^[6] Computations of p_k(TM) for concrete manifolds typically proceed via cell decompositions, which equip M with a CW structure allowing the use of the classifying space construction or recursive application of the Whitney sum formula over skeleta.^[6] Alternatively, Morse theory provides handle decompositions analogous to cell structures, where critical points induce cells, enabling calculation of characteristic classes from the attachment maps and induced bundles on handles; for example, the Pontryagin classes of complex projective spaces \mathbb{CP}^m are derived as p(\mathbb{CP}^m) = (1 + a^2)^{m+1}, with a the generator of H^2(\mathbb{CP}^m; \mathbb{Z}), using the standard Schubert cell decomposition.^[6]^[13] Pontryagin classes of manifolds can contain torsion elements, with 2-torsion particularly prominent due to their definition via Chern classes of the complexified tangent bundle, where odd-degree Chern classes c_{2k+1}(TM \otimes \mathbb{C}) have order dividing 2, implying potential 2-torsion in even-degree Pontryagin classes like p_1 = -c_2.^[6] More generally, if H^*(M; \mathbb{Z}) admits torsion, Pontryagin classes may reflect this, though examples without torsion exist; the 2-primary components of these classes further intersect with rational homotopy theory, where they contribute to the data required for realizing rational homotopy types by smooth manifolds, as the rationalized Pontryagin classes must match those of the realizing space.^[6]^[14]

Reduction Modulo 2

The Pontryagin classes p_k(E) \in H^{4k}(X; \mathbb{Z}) of a real vector bundle E over a space X reduce modulo 2 via the coefficient homomorphism H^{4k}(X; \mathbb{Z}) \to H^{4k}(X; \mathbb{Z}/2). This reduction yields the relation p_k(E) \equiv w_{2k}(E)^2 \pmod{2}, where w_{2k}(E) denotes the $2k-th Stiefel-Whitney class of E.^[15] This equality holds in the mod 2 cohomology ring and reflects the compatibility between integral and mod 2 characteristic classes for real bundles. The Stiefel-Whitney classes are interconnected with operations in mod 2 cohomology through the Wu formula, which asserts that the total Stiefel-Whitney class satisfies w(E) = \mathrm{Sq}(\nu(E)), where \mathrm{Sq} is the total Steenrod square and \nu(E) is the total Wu class of E. Expanding this relation componentwise provides recursive expressions for the even-degree Stiefel-Whitney classes; in particular, w_{2k}(E) includes a term \mathrm{Sq}^2(w_{2k-2}(E)) arising from the action on lower-degree components, linking the structure of Pontryagin classes mod 2 directly to Steenrod algebra operations.^[15] This mod 2 reduction has significant implications for the topology of manifolds. For oriented manifolds, where w_1(TM) = 0, the relation simplifies the study of even characteristic classes, but non-vanishing w_{2k}(TM) can obstruct further structures. In particular, the existence of a spin structure on a manifold requires w_2(TM) = 0, which implies p_1(TM) \equiv 0 \pmod{2}; conversely, non-spin manifolds exhibit w_2(TM) \neq 0, leading to non-trivial p_1(TM) \pmod{2} = w_2(TM)^2. For unoriented manifolds, the relation persists but interacts with the full mod 2 cohomology, highlighting differences in bundle orientability.^[15] Torsion in the integral cohomology can affect the mod 2 reduction by allowing Pontryagin classes to lie in the torsion subgroup, where a non-zero integral class may reduce to zero mod 2 if it is 2-torsion. For instance, in lens spaces, which possess 2-torsion in their cohomology rings (such as L(2,1) = \mathbb{RP}^3 or higher-dimensional analogs with \mathbb{Z}/2-torsion), the Pontryagin classes of the tangent bundle can have torsion components that vanish upon reduction, illustrating how integral structure influences the mod 2 picture.

Pontryagin Numbers

Definition and Computation

Pontryagin numbers are characteristic numbers defined for closed oriented manifolds. For a closed oriented manifold M of dimension $4m, the k-th Pontryagin number p_k[M] is given by the pairing \langle p_k(TM), [M] \rangle, where p_k(TM) \in H^{4k}(M; \mathbb{Z}) is the k-th Pontryagin class of the tangent bundle TM and [M] \in H_{4m}(M; \mathbb{Z}) is the fundamental homology class of M; this equals \int_M p_k(TM). More generally, for any monomial in Pontryagin classes p_{i_1} \cdots p_{i_r} with $4(i_1 + \cdots + i_r) = 4m, the corresponding Pontryagin number is \langle p_{i_1}(TM) \cup \cdots \cup p_{i_r}(TM), [M] \rangle \in \mathbb{Z}.^[6] These numbers are defined only for oriented manifolds, as the fundamental class [M] requires an orientation to pair with cohomology classes. Pontryagin numbers vanish for manifolds of odd dimension, since each Pontryagin class p_k has degree $4k (a multiple of 4) and thus cannot pair nontrivially with the fundamental class in odd degree. For dimensions not equal to $4m, only the relevant top-degree monomials contribute; lower-degree classes yield zero when integrated over the full manifold.^[6] Computations of Pontryagin numbers can be performed using the Atiyah–Singer index theorem, which equates the analytical index of an elliptic operator on M (such as the Dirac operator or signature operator) to an integral of characteristic classes over M, thereby providing explicit formulas for pairings like \langle p_k, [M] \rangle in terms of geometric data. Alternatively, embedding calculus techniques, as developed in the Goodwillie–Weiss tower for spaces of embeddings, allow computation of Pontryagin numbers through analysis of embedding obstructions and Haefliger invariants in high codimensions, particularly for realizing specific values in cobordism classes.^[16]^[17] Representative examples illustrate these definitions. For the 4-sphere S^4, the tangent bundle is stably trivial, so all Pontryagin classes vanish and p_1[S^4] = 0. For the complex projective plane \mathbb{CP}^2, viewed as a 4-manifold with almost complex structure, the first Pontryagin class is computed from Chern classes via the relation p_1(TM) = c_1^2(TM) - 2c_2(TM), where c_1 = 3x and c_2 = 3x^2 with x the generator of H^2(\mathbb{CP}^2; \mathbb{Z}); this yields p_1 = 3x^2 and p_1[\mathbb{CP}^2] = 3.^[6]^[6]

Integrality and Relations

Pontryagin numbers, defined as the evaluation of Pontryagin classes on the fundamental homology class of a closed oriented manifold, are always integers. This follows from the fact that the Pontryagin classes p_k \in H^{4k}(BSO(n); \mathbb{Z}) lie in integral cohomology, ensuring that their pairings with integral homology classes yield integers. The L-genus, a characteristic class constructed as a universal polynomial in the Pontryagin classes with rational coefficients, relates Pontryagin numbers to the signature of manifolds via Hirzebruch's signature theorem. The coefficients of the L-genus involve rational denominators derived from the Bernoulli numbers in the Hirzebruch polynomials, including powers of 2 and odd primes such as 3 and 5, but the overall evaluation L(M) = \sum L_k(p_1(M), \dots, p_k(M)) on a $4k-dimensional manifold is an integer because the signature \sigma(M), given by \sigma(M) = \langle L(M), [M] \rangle$, is always an integer. This integrality condition imposes constraints on the Pontryagin numbers, as the rational linear combinations must compensate for the denominators to produce integers. Pontryagin numbers exhibit bordism invariance, meaning that if two closed oriented manifolds are bordant, they have the same Pontryagin numbers. This property arises because the Pontryagin classes are natural transformations, and the evaluation map factors through the oriented bordism group \Omega_{SO}^n, where p_k[M] = 0 if M = \partial W for some oriented manifold W, as the fundamental class vanishes in bordism. Consequently, Pontryagin numbers provide complete invariants for the rationalized oriented bordism ring in certain dimensions, modulo torsion. The Hattori-Stong theorem precisely characterizes these invariants by identifying the image of the Pontryagin homomorphism in rational cohomology as the sublattice satisfying integrality conditions derived from the Adams spectral sequence. In the context of almost complex manifolds, Pontryagin numbers relate to other topological invariants such as the Euler characteristic through the Hirzebruch-Riemann-Roch theorem. Specifically, the Pontryagin classes of the real tangent bundle can be expressed in terms of the Chern classes of its complexification via p_k = (-1)^k c_{2k} + lower terms, allowing Pontryagin numbers to be rewritten as polynomials in Chern numbers. The theorem then equates the integral of the Todd genus (a polynomial in Chern classes) to the holomorphic Euler characteristic, providing a bridge between real Pontryagin invariants and complex analytic data in this setup. The Pontryagin numbers generate a torsion-free subgroup of the bordism group when mapped to integers, reflecting the absence of odd torsion in the image under the Pontryagin homomorphism. In specific cases, such as spin manifolds or dimensions congruent to 0 modulo 4, the p-adic valuations of Pontryagin numbers are bounded by those imposed by the L-genus denominators, ensuring that certain linear combinations are integrally valued without additional torsion. These aspects are fully captured by the integrality conditions of the Hattori-Stong theorem, which confirm that no further relations beyond those from Riemann-Roch and signature exist among the Pontryagin numbers.

Applications

Hirzebruch Signature Theorem

The Hirzebruch signature theorem establishes a profound connection between the topological invariant known as the signature of a manifold and its Pontryagin classes. Formulated by Friedrich Hirzebruch in the early 1950s, the theorem generalizes Rokhlin's 1952 result on the divisibility of the signature for spin 4-manifolds, extending it to higher-dimensional oriented manifolds by expressing the signature as a specific polynomial in the Pontryagin classes.^[18] For a closed oriented smooth manifold M of dimension $4m, the theorem states that the signature \sigma(M), defined as the signature of the intersection form on the middle-dimensional cohomology H^{2m}(M; \mathbb{R}), equals the evaluation of the L-genus on the fundamental class:

\sigma(M) = \langle L(TM), [M] \rangle,

where L(TM) is the total L-class of the tangent bundle TM, a characteristic class constructed from the Pontryagin classes p_i(TM). The L-genus is a multiplicative genus given by a power series in the Pontryagin classes, with the total L-class

L(p_1, p_2, \dots) = \prod_{i=1}^n \frac{x_i/2}{\tanh(x_i/2)},

where the p_i are formal power series roots related to the eigenvalues of the curvature. The first few components are the homogeneous polynomials

L_1 = \frac{p_1}{3}, \quad L_2 = \frac{7p_2 - p_1^2}{45},

and higher-degree terms follow similarly as rational polynomials ensuring integrality on manifolds. For example, on a 4-manifold, the theorem simplifies to \sigma(M) = \langle p_1/3, [M] \rangle.^[19] A modern proof of the theorem utilizes the Atiyah-Singer index theorem applied to the signature operator on M. The signature operator D = d + d^* acts on the space of differential forms \Omega^*(M), graded into even and odd degrees, with its index \operatorname{ind}(D) = \dim \ker D^+ - \dim \ker D^- equaling \sigma(M) by Hodge theory, as the kernel corresponds to harmonic forms isomorphic to de Rham cohomology. The Atiyah-Singer theorem computes this index as the integral of the local index density, which for the signature operator yields precisely the L-genus: \operatorname{ind}(D) = \int_M \hat{A}(TM) \cdot \operatorname{ch}(S), but specialized to the signature complex, it reduces to \int_M L(TM), confirming the topological expression. This analytic approach, developed in 1963, provides an alternative to Hirzebruch's original topological proof via oriented cobordism and multiplicative sequences.^[20]

Examples on Complex Surfaces

For a complex surface S, the first Pontryagin class of its tangent bundle is given by p_1(TS) = c_1^2 - 2c_2, where c_1 and c_2 denote the first and second Chern classes of TS.^[6] This relation arises from the underlying real vector bundle structure of the complex tangent bundle. The Noether formula further connects these invariants to the holomorphic Euler characteristic: \chi(\mathcal{O}_S) = \frac{1}{12}(c_1^2 + c_2).^[21] A prominent example is the quartic K3 surface, realized as a smooth hypersurface of degree 4 in \mathbb{CP}^3. For any K3 surface, the first Chern class vanishes, c_1 = 0, while the second Chern class integrates to c_2 = 24, yielding \chi(\mathcal{O}_S) = 2 via the Noether formula. Consequently, p_1 = -2 \times 24 = -48, and the Pontryagin number \langle p_1, [S] \rangle = -48. This implies a signature of -16 for the intersection form on H^2(S; \mathbb{Z}), reflecting the surface's topological rigidity. Enriques surfaces provide another illustrative case, with Chern numbers c_1^2 = 0 and c_2 = 12, consistent with \chi(\mathcal{O}_S) = 1. The relation then gives p_1 = -24, leading to a signature of -8. Unlike K3 surfaces, the canonical bundle K_S is nontrivial but 2-torsion, $2K_S \cong \mathcal{O}_S, which influences adjunction formulas for embedded curves: for a curve C \subset S, the genus is g(C) = 1 + \frac{1}{2}(C^2 + K_S \cdot C). This torsion property geometrically interprets the vanishing c_1^2 and ties Pontryagin classes to the surface's minimal model. Elliptic surfaces, fibered over a base curve with elliptic fibers, exhibit Pontryagin classes modulated by the fibration structure and multiple fibers. For a minimal elliptic surface S \to B of canonical bundle degree d = \langle c_1^2, [S] \rangle, the Noether formula yields \chi(\mathcal{O}_S) = d/12, with p_1 computable via the Chern relation once fiber contributions are accounted for. Geometric insights arise from adjunction on sections and fibers, where the canonical bundle restricts to the fiber's structure, linking p_1 to logarithmic transforms and the surface's Kodaira dimension.

Role in Surgery Theory

In surgery theory, Pontryagin classes and the associated Pontryagin numbers provide key obstructions to the existence of normal maps between manifolds that can be transformed into homotopy equivalences via surgery. This role is central to the classification of high-dimensional manifolds up to diffeomorphism or homeomorphism, particularly in the context of oriented manifolds where these invariants detect whether a manifold can be surgically modified to match a target space.^[22]^[23] The Kervaire-Milnor surgery exact sequence formalizes this by relating the structure set S(X), which classifies manifolds homotopy equivalent to a Poincaré complex X, to the group of normal invariants N(X) and the algebraic surgery obstruction groups L_n(\mathbb{Z}[\pi_1(X)]):

S(X) \to N(X) \to L_n(\mathbb{Z}[\pi_1(X)])

Here, elements of N(X) correspond to stable normal bundle maps over homotopy equivalences to X, and the connecting homomorphism to L_n assigns a surgery obstruction that vanishes precisely when the normal map is cobordant to a homotopy equivalence. For simply connected oriented manifolds of dimension n \geq 5, this obstruction reduces to the vector of Pontryagin numbers of the source manifold M, specifically requiring that \langle p_i(M), [M] \rangle = 0 for all i when X is a sphere, as spheres have vanishing Pontryagin classes. Non-vanishing Pontryagin numbers thus obstruct the surgery, preventing M from bounding a contractible manifold.^[22]^[23]^[24] These obstructions connect directly to the algebraic surgery groups L_*, which classify metabolic quadratic forms over rings like \mathbb{Z}[\pi]. In the oriented case with trivial fundamental group, L_{4k}(\mathbb{Z}) \cong \mathbb{Z} is generated by the signature (linked topologically to Pontryagin numbers), while L_{4k+2}(\mathbb{Z}) \cong \mathbb{Z}/2 is generated by the Arf-Kervaire invariant, a mod-2 quadratic refinement that incorporates Pontryagin classes reduced modulo 2 via their relation to Stiefel-Whitney classes. For non-simply connected manifolds, the groups L_n(\mathbb{Z}[\pi]) incorporate twisted coefficients, but Pontryagin numbers still appear in the unoriented or oriented components as integral invariants of the quadratic forms.^[23]^[24] Wall's finiteness theorem establishes that, for a compact oriented manifold M^n (n \geq 5) with finite fundamental group, the structure set S(M) is finite, relying on the finiteness of the Whitehead group \mathrm{Wh}(\pi_1(M)) and the surgery groups L_n(\mathbb{Z}[\pi_1(M)]), where Pontryagin classes bound the possible normal invariants in [M, G/O]. Complementing this, Wall's periodicity theorem asserts a 4-fold periodicity in the surgery obstruction groups, L_{n+4}(\mathbb{Z}[\pi]) \cong L_n(\mathbb{Z}[\pi]) \oplus L_4(\mathbb{Z}), allowing computations in higher dimensions to reduce to low-dimensional cases involving Pontryagin classes p_k for k \leq n/4; this periodicity facilitates the use of Pontryagin numbers as recurring obstructions in oriented surgery sequences.^[23]^[25] A concrete example of these obstructions arises in the classification of homotopy spheres: in dimension 7, the 28 exotic 7-spheres exist, but certain candidates are ruled out because their bounding parallelizable 8-manifolds would require non-zero Pontryagin numbers \langle p_1, [W] \rangle \neq 0, which contradict the vanishing required for the bounding manifold in the surgery sequence. Similarly, no smooth structure exists on certain PL homotopy spheres in higher dimensions, such as the Kervaire manifold in dimension 10, due to mismatched Pontryagin numbers obstructing the normal map to the standard sphere.^[22]^[24]

Generalizations

To Stable Normal Bundles

In the context of embedding theory, the stable normal bundle \nu^s(M) of an n-dimensional manifold M is defined as the stabilization of the normal bundle \nu arising from an embedding i: M \hookrightarrow \mathbb{R}^{n+k} for sufficiently large k, specifically \nu^s(M) = \nu \oplus \varepsilon^l where \varepsilon^l is the trivial line bundle repeated l times and l is chosen large enough for stability.^[1] Since the tangent bundle satisfies TM \oplus \nu = \varepsilon^{n+k}, it follows in K-theory that [\nu] = -[TM], and the Pontryagin classes, being stable under addition of trivial bundles, satisfy p(\nu^s(M)) = p(\nu) = p(-TM) = p(TM), as the Pontryagin classes of a real vector bundle E and its opposite -E coincide via complexification, where p_i(E) = (-1)^i c_{2i}(E_\mathbb{C}) and Chern classes are invariant under orientation reversal.^[1] The Haefliger-Hirsch theorem establishes that, in the appropriate dimension range, embeddings of compact manifolds are classified up to isotopy by homotopy classes of stable normal maps, which are maps M \to [BO](/page/Bo) representing the stable normal bundle \nu^s(M); these classes serve as primary invariants, with the Pontryagin classes p_i(\nu^s(M)) providing cohomological obstructions that distinguish non-isotopic embeddings. Specifically, for an n-manifold with $0 \leq \kappa \leq (n-4)/2, the isotopy classes of embeddings into \mathbb{R}^{2n - \kappa} correspond bijectively to certain homotopy classes related to the stable normal bundle, where the Pontryagin classes must match those of the classifying map to ensure compatibility.^[26] Computations of these invariants in the metastable range—typically where the codimension k > (n-3)/2—rely on immersion theory, reducing embedding problems to those of immersions via normal vector fields; here, the rational Pontryagin classes of \nu^s(M) determine the stable equivalence class of the bundle, allowing explicit calculation via pullbacks from the Grassmannian Gr_n(\mathbb{R}^{n+k}), as every closed n-manifold immerses in \mathbb{R}^{2n - \alpha(n)} (with \alpha(n) the number of 1's in the binary expansion of n) if the relevant Stiefel-Whitney classes vanish, and Pontryagin classes refine the rational classification.^[26] For example, the Pontryagin classes can be computed as p_i(\nu^s(M)) = f^* p_i(\gamma^{n+k}) for a classifying map f: M \to BO(n+k), yielding invariants that are independent of the specific high-dimensional embedding.^[1] In contrast to the stable regime, unstable cases in low dimensions exhibit additional obstructions where Pontryagin classes alone do not suffice for classification; for instance, the real projective plane \mathbb{RP}^2 fails to immerse in \mathbb{R}^3 despite its stable normal bundle having trivial Pontryagin classes rationally, due to nontrivial homotopy groups of the Stiefel manifold V_{2,3}, whereas in dimensions n \geq 4, such low-dimensional phenomena resolve and stable invariants like Pontryagin classes dominate.^[26] This distinction highlights how, below the metastable threshold (e.g., codimension k \leq (n-3)/2), local self-intersections and unstable bundle phenomena introduce complexities not captured by stable Pontryagin classes.

Equivariant Pontryagin Classes

Equivariant Pontryagin classes generalize the classical Pontryagin classes to the setting of vector bundles equipped with compatible group actions. For a topological group G acting on a space X and a G-equivariant real vector bundle E \to X of rank n, the k-th equivariant Pontryagin class p_k^G(E) is defined as an element of the equivariant cohomology group H_G^{4k}(X; \mathbb{Z}).^[27] This cohomology is computed via the Borel construction, where H_G^*(X; \mathbb{Z}) \cong H^*(EG \times_G X; \mathbb{Z}), with EG the universal G-space. The bundle E corresponds to a G-equivariant classifying map f: X \to EG \times_G BO(n), and p_k^G(E) = f^* p_k, where p_k \in H^{4k}(BO(n); \mathbb{Z}) is the ordinary universal Pontryagin class.^[27] The construction ensures naturality under equivariant bundle maps, preserving the core properties of nonequivariant Pontryagin classes such as multiplicativity and stability under Whitney sums. For torus actions, where G = T is a torus, explicit computations often rely on localization techniques in equivariant cohomology. Fixed-point formulas arise from the Atiyah-Bott-Berline-Vergne localization theorem, which expresses integrals of equivariant classes as sums over fixed-point components: for a T-invariant form \omega on X, \int_X \omega = \sum_C \frac{\int_C \omega|_C}{e_T(N_C)}, where C runs over connected components of the fixed-point set X^T, and e_T(N_C) is the equivariant Euler class of the normal bundle to C. Applied to Pontryagin classes on spaces like real Grassmannians G_k(\mathbb{R}^n) with the standard torus action, the localized classes at fixed points S (subsets of basis indices) satisfy p_k^T|_S = \prod_{i \in S} (1 + \alpha_i^2)^{k} or related products involving equivariant parameters \alpha_i, enabling explicit evaluation. The relation to nonequivariant Pontryagin classes is captured by the forgetful map H_G^*(X; \mathbb{Z}) \to H^*(X; \mathbb{Z}), which sends p_k^G(E) to the ordinary p_k(E), reflecting the underlying bundle structure when the action is trivialized. More deeply, the Serre spectral sequence of the fibration X \to EG \times_G X \to [BG](/page/BG) converges to H_G^*(X; \mathbb{Z}), with E_2^{p,q} = H^p([BG](/page/BG); H^q(X; \mathbb{Z})); this sequence relates equivariant classes to twists of ordinary cohomology by the cohomology of [BG](/page/BG), providing a tool to compute restrictions and extensions. Applications of equivariant Pontryagin classes extend to representation theory, where they inform the structure of equivariant cohomology rings for representation spaces, such as Grassmannians, yielding bases in terms of symmetric polynomials and facilitating computations of characteristic numbers via localization. A prominent use is in the G-signature theorem, which decomposes the signature of a G-manifold into contributions from fixed-point sets, weighted by idempotents in the rational group ring \mathbb{Q}[G]; equivariant Pontryagin classes enter via their evaluations on orbit spaces and symmetric products, enabling obstructions to group actions and classifications of transformation groups. Zagier's seminal work applies this framework to derive constraints on finite group actions from G-signatures computed using these classes.^[27]

In Cobordism Rings

The oriented cobordism ring \Omega^{SO}_* consists of cobordism classes of compact oriented smooth manifolds, graded by dimension, with addition induced by disjoint union and multiplication by Cartesian product, making it a commutative graded ring concentrated in even degrees.^[28] Pontryagin classes p_i \in H^{4i}(-; \mathbb{Z}) of the tangent bundle provide characteristic classes for oriented manifolds that are natural under cobordism, inducing ring homomorphisms from \Omega^{SO}_* to cohomology rings via pullback along classifying maps for the stable tangent bundle. Specifically, for a class [M] \in \Omega^{SO}_{4k}, the Pontryagin numbers \int_M p_{i_1}^{a_1} \cdots p_{i_r}^{a_r} (where the multi-index corresponds to a partition of k) are well-defined integers independent of the representative manifold and define additive group homomorphisms \Omega^{SO}_{4k} \to \mathbb{Z}. These extend multiplicatively to ring maps on the graded components due to the product formula for Pontryagin classes under Cartesian product: p(TM \times TN) = p(TM) \cup p(TN).^[28]^[29] Thom proved that the Pontryagin numbers yield p(k) linearly independent homomorphisms, where p(k) is the number of integer partitions of k, implying \mathrm{rank}(\Omega^{SO}_{4k} \otimes \mathbb{Q}) = p(k); moreover, the generators of \Omega^{SO}_{4k} \otimes \mathbb{Q} can be realized by products of complex projective spaces \mathbb{CP}^2, \mathbb{CP}^4, \dots, \mathbb{CP}^{2k}, each with non-vanishing Pontryagin numbers that span the dual basis.^[28]^[30] Conner and Floyd established that \Omega^{SO}_* \otimes \mathbb{Q} is isomorphic as a graded ring to the polynomial algebra \mathbb{Q}[b_4, b_8, b_{12}, \dots] on generators b_{4i} of degree $4i, where the b_{4i} are the images of [\mathbb{CP}^{2i}] under the Hurewicz map to homology, and the multiplication is determined by the Pontryagin classes of these generators via the universal property of the ring homomorphisms induced by integrals of symmetric polynomials in the p_j. This structure reflects how the Pontryagin classes embed the cobordism ring into the algebra generated by cohomology operations.^[30]^[31]

References

[1]
[PDF] Version 2.2, November 2017 Allen Hatcher Copyright c 2003 by ...
The easiest way to define Pontryagin classes is in terms of Chern classes rather ... Here is how Pontryagin classes are related to Stiefel-Whitney and Euler ...
[2]
[PDF] Lectures on Algebraic Topology - MIT Mathematics
74 Closing the Chern circle, and Pontryagin classes . ... An ultimate goal of algebraic topology is to find means to compute the set of homotopy classes.
[3]
[PDF] CHARACTERISTIC CLASSES
In 1942 LEV PONTRJAGIN of Moscow University began to study the homology of Grassmann manifolds, using a cell subdivision due to Charles. Ehresmann. This enabled ...
[4]
[PDF] 18.906 Algebraic Topology II Chapter 5 - MIT OpenCourseWare
The sign is there in the definition of the Pontryagin classes so that the following important relation is satisfied. Lemma 36.4. For any oriented 2k-plane ...
[5]
L. S. Pontryagin, “Characteristic cycles on differentiable manifolds ...
This article is cited in 2 scientific papers (total in 2 papers) Characteristic cycles on differentiable manifolds LS Pontryagin Full-text PDF (6108 kB)
[6]
[PDF] Characteristic Classes
Sep 28, 2021 · • John Milnor and James Stasheff, Characteristic Classes. – Available ... We will now give an inductive definition of characteristic classes for a ...
[7]
[PDF] Vector Bundles. Characteristic classes. Cobordism. Applications
May 3, 2018 · Stiefel-Whitney classes of O(n)-bundles enjoy similar properties as the Chern classes. Proposition 3.5. The Stiefel-Whitney classes satisfy the ...<|control11|><|separator|>
[8]
[PDF] 1. Pontryagin Classes
Before we start defining Pontryagin classes, we need a few more lemmas concerning Chern classes. Definition 1.1. The complexification of a real vector bundle V ...
[9]
https://doi.org/10.1201/9780367813758
[10]
https://doi.org/10.1007/978-1-4757-3951-0
[11]
https://doi.org/10.1090/mmono/201
[12]
[PDF] LECTURE 10 Characteristic classes - webspace.science.uu.nl
The Pontryagin class has the same property as the Chern class: for any two ... The Banach axioms. 42. 3.3. Invariance axiom. 43. 3.4. Density axioms. 44. 3.5 ...
[13]
Pontryagin classes of a tensor product of bundles
May 16, 2015 · Therefore, you can obtain a formula for the ith Pontryagin class ... Formula for the Stiefel-Whitney classes of a tensor product · 2 · Are ...
[14]
Computation of characteristic classes of a manifold from a ...
Abstract. This paper is devoted to the well-known problem of computing the. Stiefel–Whitney classes and the Pontryagin classes of a manifold from a given ...
[15]
[PDF] on the characterization of rational homotopy types and chern ...
Pontryagin [Po42] studied the homology of these universal spaces, identifying the generators of the integral lattice in rational (co)homology now known as ...<|control11|><|separator|>
[16]
http://math.uchicago.edu/~shmuel/tom-readings/ASI.pdf
[17]
[PDF] The Index of Elliptic Operators: I - MF Atiyah, IM Singer
Feb 25, 2002 · This is the first of a series of papers which will be devoted to a study of the index of elliptic operators on compact manifolds. The main ...
[18]
Rational Pontryagin classes and functor calculus - EMS Press
In this paper we reformulate the hypothesis as a statement in differential topology, and also in a functor calculus setting. Keywords. Pontryagin classes, ...
[19]
1953 - Ideas | Institute for Advanced Study
Friedrich Hirzebruch's 1954 paper (shown here), which announced the ... theorem to the signature theorem that Hirzebruch had proved six months earlier.
[20]
[PDF] The signature theorem: reminiscences and recreation
[17] Hirzebruch, F.: Pontrjagin classes of rational homology manifolds and the signature of some affine hypersurfaces. Liverpool Symposium on singularities ...Missing: Pontryagin | Show results with:Pontryagin
[21]
[PDF] the atiyah-singer index theorem - Berkeley Math
Proof Sketch. ... We will now realize the Chern-Gauss-Bonnet Theorem and Hirzebruch Signature. Theorem as consequences of the Atiyah-Singer index theorem.
[22]
https://webhomes.maths.ed.ac.uk/~v1ranick/papers/kervmiln.pdf
[23]
[PDF] Groups of homotopy spheres I.
GROUPS OF HOMOTOPY SPHERES: I. BY MICHEL A. KERVAIRE AND JOHN W. MILNOR ... KERVAIRE AND MILNOR a tubular neighborhood of this arc is diffeomorphic to R ...
[24]
[PDF] SURGERY ON COMPACT MANIFOLDS
Page 1. SURGERY ON. COMPACT MANIFOLDS. C. T. C. Wall. Second Edition. Edited by A. A. Ranicki. Page 2. ii. Prof. C.T.C. Wall, F.R.S.. Dept. of Mathematical ...
[25]
None
Below is a merged summary of Pontryagin classes and numbers in surgery theory, consolidating all information from the provided segments into a comprehensive response. To retain maximum detail, I will use a structured format with sections and tables where appropriate, ensuring all key points, theorems, citations, and URLs are included. The response avoids redundancy while preserving the richness of the content.
[26]
[PDF] Surveys on Surgery Theory : Volume 1 Papers dedicated to CTC Wall
Wall, one of the leaders of the founding generation of the subject, led us to reflect on the extraordinary accomplish- ments of surgery theory, and its current ...
[27]
[PDF] Bundles, Homotopy, and Manifolds - Mathematics
stable isomorphism classes of vector bundles SV ect(X) is isomorphic to the ... space” for normal bundles of n-manifolds, and proved the following theorem.
[28]
Equivariant Pontrjagin Classes and Applications to Orbit Spaces
Authors. Don Bernard Zagier ; Book Title Equivariant Pontrjagin Classes and Applications to Orbit Spaces ; Book Subtitle Applications of the G-signature Theorem ...
[29]
[PDF] 1. The Oriented Cobordism Ring
(Pontryagin) If M and M0 are oriented cobordant 4k manifolds then they have the same Pontryagin numbers. Proof. Since M t −M0 is the oriented boundary of a 4k + ...
[30]
[PDF] Thom Spaces and the Oriented Cobordism Ring
May 20, 2020 · An n-dimensional manifold M is an oriented boundary if and only if all. Pontrjagin numbers and all Stiefel-Whitney classes vanish. Branko Juran.
[31]
[PDF] The oriented cobordism ring
The computation of π∗MSO splits into three parts: rational data, odd primary data, and even primary data. In §3, we investigate the rational data. This part ...
[32]
Vertical genera - ScienceDirect
The genera considered here are built from certain generalised Pontryagin and Chern classes as maps from bordism cohomology to the singular cohomology of X .