Fact-checked by Grok 2 weeks ago

Characteristic class

In algebraic topology, characteristic classes are natural transformations that assign cohomology classes to vector bundles or principal bundles over a base space, serving as topological invariants that capture the bundle's "twisting" or global structure.^[1]^[2] These classes arise from the classifying space construction: for a principal G-bundle P \to M, a classifying map f: M \to BG (where BG is the classifying space of the Lie group G) pulls back universal cohomology classes from H^*(BG; A) to H^*(M; A), yielding invariants natural under bundle pullbacks.^[3] They originated in mid-20th-century differential topology to reconcile local linear approximations of manifolds with their global topological properties, providing obstructions to phenomena like orientability or the existence of nowhere-zero sections.^[2] Prominent examples include the Stiefel-Whitney classes for real vector bundles, valued in mod-2 cohomology and detecting orientability and immersibility; the Chern classes for complex vector bundles, defined integrally via projective bundles and satisfying axioms like additivity under Whitney sums (c(E \oplus F) = c(E) \cup c(F)); the Pontryagin classes for real bundles, quadratic in terms of Chern classes; and the Euler class for oriented bundles, measuring self-intersections in zero sections.^[1]^[3] Characteristic classes bridge geometry and algebra by enabling computations of manifold invariants, such as cobordism groups through characteristic numbers (integrals of products of classes over the fundamental class), and play key roles in index theorems, K-theory, and the study of bundle obstructions like spin structures.^[1]^[2] Their computation often relies on connections and curvature forms on principal bundles, which quantify parallel transport failures and relate to differential forms via Chern-Weil theory.^[3]

Fundamentals

Definition

In algebraic topology, a characteristic class assigns to each vector bundle E over a paracompact base space M an element of the cohomology group H^k(M; A) for some integer k \geq 0 and coefficient ring A, depending only on the isomorphism class of E, and satisfying naturality under pullbacks: for any continuous map \psi: N \to M, the characteristic class of the pullback bundle \psi^* E over N is the pullback \psi^* of the characteristic class of E over M.^[4] This construction extends to families of such classes forming elements of the cohomology ring H^*(M; A).^[4] More abstractly, a characteristic class of degree k for principal G-bundles (or the associated vector bundles) is a natural transformation from the functor assigning to each space the isomorphism classes of principal G-bundles over it to the cohomology functor H^k(-; A), and by the Yoneda lemma or representability of cohomology, this corresponds uniquely to a universal class in H^k(BG; A), where BG is the classifying space of the group G.^[4] For a given principal G-bundle P \to M classified by a map f: M \to BG, the characteristic class on M is the pullback f^* of the universal class.^[5] These classes serve as topological invariants that partially classify vector bundles up to isomorphism: two bundles over M are isomorphic if and only if their classifying maps to the Grassmannian G_n(R^\infty) or BG are homotopic, and the characteristic classes capture this via the induced maps on cohomology.^[4] The universal characteristic classes reside in the cohomology of the classifying space or Grassmannian, providing a complete set of invariants in many cases.^[5] For example, the k-th Stiefel-Whitney class w_k(E) of a real vector bundle E is an element of H^k(M; \mathbb{Z}/2\mathbb{Z}), while the k-th Chern class c_k(E) of a complex vector bundle is in H^{2k}(M; \mathbb{Z}).^[4]

Historical development

The theory of characteristic classes originated in the mid-1930s with the independent and nearly simultaneous contributions of Eduard Stiefel and Hassler Whitney, who introduced what are now known as the Stiefel-Whitney classes for real vector bundles. Stiefel's work in 1935 focused on the topology of Stiefel manifolds and sphere bundles, motivated by problems in classifying immersions of manifolds into Euclidean spaces and identifying obstructions to such embeddings.^[4] Whitney, in his 1935 paper on products in a complex, developed these classes as cohomology invariants associated to the tangent bundles of smooth manifolds, providing tools to study the embedding and immersion properties of manifolds. These classes, defined modulo 2, captured essential topological features and laid the groundwork for later generalizations. In the 1940s, Shiing-Shen Chern extended the concept to complex vector bundles, defining the Chern classes through a differential geometric approach that linked them directly to the curvature of connections on principal bundles. Chern's seminal 1946 paper integrated topological invariants with geometric structures, showing how these classes arise from the Chern-Weil homomorphism applied to curvature forms, thus bridging algebraic topology and differential geometry.^[6] This formulation not only generalized the Stiefel-Whitney classes but also provided a means to compute them via integral cohomology, influencing subsequent developments in bundle theory. Meanwhile, Lev Pontryagin introduced his namesake classes around 1940-1942 for real vector bundles, motivated by the study of manifolds that bound other manifolds, where these classes vanish as obstructions.^[7] The 1950s saw significant refinements and integrations of characteristic classes into broader topological frameworks. Friedrich Hirzebruch's 1954 Riemann-Roch theorem generalized the classical Riemann-Roch formula to higher-dimensional algebraic varieties, expressing the holomorphic Euler characteristic in terms of Todd and Chern classes, thereby connecting characteristic classes to index theory and arithmetic genera.^[8] John Milnor's work in the same decade on stable homotopy groups and cobordism theory further refined the understanding of these classes, particularly through their role in stable equivalence of bundles and computations in the stable range.^[4] Pontryagin classes were also formalized more rigorously during this period as quadratic refinements of Chern classes for real bundles. Later advancements in the 1960s and beyond highlighted the deep connections between characteristic classes and analysis. The 1963 Atiyah-Singer index theorem, announced by Michael Atiyah and Isadore Singer, equated the analytic index of elliptic operators on compact manifolds to a topological index expressed via characteristic classes of the tangent bundle and associated bundles, unifying differential operators with global invariants. In the 1980s, Simon Donaldson's introduction of gauge-theoretic invariants for four-manifolds relied heavily on Pontryagin classes to construct polynomial invariants distinguishing smooth structures, with applications extending to Donaldson-Witten theory in quantum field theory. These developments underscored the enduring impact of characteristic classes across topology, geometry, and physics.

Properties

Stability

In algebraic topology, two vector bundles E and F over the same base space are stably equivalent if there exist non-negative integers k and m such that E \oplus \epsilon^k \cong F \oplus \epsilon^m, where \epsilon denotes the trivial line bundle of rank 1.^[4] This equivalence captures the idea that bundles differing only by trivial summands represent the same stable class, which is fundamental to the theory of characteristic classes.^[4] A characteristic class c_k is stable if it remains unchanged when a trivial bundle is added, meaning c_k(E \oplus \epsilon) = c_k(E) for any vector bundle E.^[4] This invariance implies a multiplicative structure under the Whitney sum operation: the total characteristic class satisfies c(E \oplus F) = c(E) \cdot c(F), or more precisely, c_k(E \oplus F) = \sum_{i+j=k} c_i(E) c_j(F).^[4] For instance, Stiefel-Whitney classes exhibit this stability, as w_i(\xi \oplus \epsilon) = w_i(\xi) for real bundles \xi.^[4] Stability manifests in the stable range, where characteristic classes of a bundle E of rank n become constant after adding sufficiently many trivial line bundles, typically for dimensions beyond a certain threshold related to the base space.^[4] Classes such as Chern classes for complex bundles and Pontryagin classes for real bundles are stable in this sense, satisfying c_k(\omega \oplus \phi) = c_k(\omega) under trivial addition and exhibiting multiplicativity.^[4] In contrast, the Euler class is unstable, as it does not preserve invariance under stabilization.^[4] This stable behavior is intimately connected to the infinite Grassmannian \mathrm{Gr}_\infty, the direct limit of Grassmannians \mathrm{Gr}_n as n \to \infty, which classifies stable vector bundles.^[4] The cohomology ring H^*(\mathrm{Gr}_\infty; \mathbb{Z}/2) for real bundles is a polynomial algebra generated by the universal Stiefel-Whitney classes, while H^*(\mathrm{Gr}_\infty(\mathbb{C}); \mathbb{Z}) is generated by Chern classes, providing a universal home for these stable invariants.^[4]

Naturality and functoriality

Characteristic classes satisfy a naturality axiom, which ensures their compatibility with maps between bundles and their base spaces. Specifically, for a vector bundle map f: E \to F covering a continuous base map \phi: M \to N, where E \to M and F \to N, the induced pullback \phi^*: H^*(N; R) \to H^*(M; R) in cohomology with coefficients in a ring R satisfies \phi^* c_k(F) = c_k(E) for each characteristic class c_k, or equivalently, c_k(f^* F) = \phi^* c_k(F).^[9] This property follows from the definition of characteristic classes as natural transformations from the functor of vector bundles to the functor of cohomology groups.^[10] The functorial nature of characteristic classes arises directly from their role as natural transformations, preserving structure under operations such as pullbacks along base maps and, in appropriate settings, tensor products of bundles. For instance, under pullback, the classes transform covariantly, maintaining their invariant properties across different spaces. This functoriality implies that characteristic classes serve as stable invariants that respect the categorical structure of bundle morphisms and base space maps.^[9] As a special case, naturality holds when adding trivial bundles, relating to stability properties.^[10] A key implication of naturality and functoriality is their role in classifying vector bundles up to isomorphism. When the characteristic classes generate the cohomology ring of the classifying space (such as BO for real bundles or BU for complex bundles), they fully distinguish non-isomorphic bundles by providing complete topological invariants via the pullback of universal classes.^[10] Naturality extends across various coefficient rings and cohomology theories. For example, Stiefel-Whitney classes use \mathbb{Z}/2-coefficients in mod 2 cohomology, Chern classes use integer or rational coefficients in ordinary cohomology, and Pontryagin classes often employ rational coefficients; in each case, the pullback compatibility holds. This extends to generalized cohomology theories, where characteristic classes act as natural transformations preserving the functorial behavior under bundle maps and base pullbacks.^[9]

Constructions

Axiomatic characterization

Characteristic classes can be characterized axiomatically as systems of cohomology classes associated to vector bundles that satisfy certain natural and algebraic properties. For a sequence of characteristic classes c_k taking values in the cohomology ring H^*(X; \mathbb{Z}) (or appropriate coefficients) of the base space X, the axioms typically include: (1) naturality under pullbacks, meaning that for any bundle map f: E \to F covering a map p: X \to Y, c_k(E) = p^* c_k(F); (2) normalization on universal bundles, such that the classes c_k on the universal bundle over the classifying space BG generate the cohomology ring H^k(BG) in the appropriate degree; and (3) multiplicativity under Whitney sum, where the total characteristic class satisfies c(E \oplus F) = c(E) \cup c(F) in the cohomology ring, implying c_k(E \oplus F) = \sum_{i+j=k} c_i(E) \cup c_j(F).^[11] These axioms ensure that the characteristic classes behave functorially with respect to bundle morphisms and decompositions, capturing essential topological features of the bundles. A key result is the uniqueness theorem, which states that any system of classes satisfying these axioms is uniquely determined by the universal characteristic classes on BG, as the pullback along the classifying map f: X \to BG for a bundle E \to X recovers all classes via c_k(E) = f^* c_k(\gamma), where \gamma is the universal bundle.^[11] The axiomatic framework extends naturally to generalized cohomology theories, such as K-theory, where characteristic classes correspond to elements in the associated cohomology rings that satisfy analogous naturality, normalization, and multiplicativity axioms; for instance, in complex K-theory, the Chern character provides such a map from K^0(X) to ordinary cohomology.^[12] In the complex case, the splitting principle further simplifies computations by allowing any complex vector bundle to be pulled back to a flag manifold where it decomposes as a sum of line bundles, reducing higher Chern classes to symmetric polynomials in the first Chern classes of these line bundles, consistent with the axioms.^[9]

Via classifying spaces

In algebraic topology, the classifying space BG of a topological group G is defined as the base space of a universal principal G-bundle EG \to BG, where the total space EG is contractible.^[5] This construction ensures that BG classifies principal G-bundles up to isomorphism: any principal G-bundle P \to M over a paracompact base M is isomorphic to the pullback f^* EG \to M along a classifying map f: M \to BG.^[4] For vector bundles, relevant groups include the orthogonal group O(n) for real bundles and the unitary group U(n) for complex bundles, with classifying spaces BO(n) and BU(n) realized as infinite Grassmann manifolds G_n(\mathbb{R}^\infty) and G_n(\mathbb{C}^\infty), respectively.^[4] Characteristic classes arise naturally from this setup. A characteristic class of degree k for G-bundles is specified by a universal cohomology class u_k \in H^k(BG; A) with coefficients in a ring A; for a bundle classified by f: M \to BG, the corresponding class is the pullback c_k(P) = f^* u_k \in H^k(M; A).^[4] In the stable range, one considers the direct limits BO = \varinjlim BO(n) and BU = \varinjlim BU(n), which classify stable vector bundles. The cohomology ring H^*(BO; \mathbb{Z}/2\mathbb{Z}) is a polynomial algebra generated by the Stiefel-Whitney classes w_i of degree i, while H^*(BU; \mathbb{Z}) is a polynomial algebra generated by the Chern classes c_i of degree $2i.^[4] These universal classes are pulled back from the canonical (universal) bundles over BO and BU, ensuring the construction satisfies naturality under bundle maps.^[4] A deeper geometric realization of these classes involves the transgression map in the Serre spectral sequence of the universal fibration G \to EG \to BG. Since EG is contractible, H^*(EG; A) = A in degree 0 and vanishes otherwise, so the spectral sequence E_2^{p,q} = H^p(BG; H^q(G; A)) converges to H^*(EG; A) and collapses accordingly.^[13] The transgression is the differential d_{q+1}: E_2^{0,q} \to E_2^{q+1,0}, which maps nontrivial classes in the fiber cohomology H^q(G; A) (permanent cycles on the edge) to classes in H^{q+1}(BG; A) on the base.^[13] For example, primitive elements x_i in H^{2r_i - 1}(G; k) (with k a field) transgress to generators y_i in H^{2r_i}(BG; k), yielding the polynomial structure of H^*(BG; k).^[13] This mechanism produces the universal characteristic classes as transgressive elements, linking the topology of the structure group G to invariants on BG.^[13]

Specific examples

Stiefel-Whitney classes

Stiefel-Whitney classes are characteristic classes associated to real vector bundles, taking values in the mod 2 cohomology of the base space. For an n-dimensional real vector bundle E over a paracompact space M, the individual Stiefel-Whitney classes are elements w_i(E) \in H^i(M; \mathbb{Z}/2) for i = 0, 1, \dots, n, with w_0(E) = 1 and w_i(E) = 0 for i > n. The first Stiefel-Whitney class w_1(E) detects the orientability of the bundle, vanishing if and only if E is orientable, while the top class w_n(E) coincides with the Euler class of E reduced modulo 2. The total Stiefel-Whitney class is the formal sum w(E) = 1 + w_1(E) + \cdots + w_n(E) \in H^*(M; \mathbb{Z}/2).^[11] These classes satisfy key algebraic properties that reflect operations on vector bundles. Under the Whitney sum of bundles over the same base, the total classes multiply via the cup product: w(E \oplus F) = w(E) \cup w(F). These relations follow from the naturality of the classes under bundle maps and the structure of the cohomology ring of Grassmannians.^[11] For manifolds, the Stiefel-Whitney classes of the tangent bundle TM are related to the cohomology operations via Wu's formula. Specifically, the total Stiefel-Whitney class satisfies w(TM) = \Phi^{-1} \mathrm{Sq}(u), where u \in H^*(M; \mathbb{Z}/2) is the Thom class of the normal bundle (or fundamental class in the manifold case), \Phi is the Thom isomorphism, and \mathrm{Sq} denotes the total Steenrod square operation. Equivalently, in terms of Wu classes v_k \in H^k(M; \mathbb{Z}/2) defined by \mathrm{Sq}(x) = v \cup x for all x \in H^*(M; \mathbb{Z}/2), the formula expresses w_k(TM) = \sum_{i=0}^k \mathrm{Sq}^i(v_{k-i}). This relation, established by Wu Wen-Tsün, links the topology of the manifold directly to its Stiefel-Whitney classes and provides a computational tool for determining these classes from the action of Steenrod operations.^[11] Stiefel-Whitney classes play a crucial role in embedding and immersion theory. They provide necessary conditions for the existence of immersions into Euclidean spaces of given codimension, arising from the requirement that the formal inverse of the total class w(TM) has vanishing coefficients in degrees exceeding the codimension. All Stiefel-Whitney classes are stable under addition of trivial bundles, meaning w_i(E \oplus \epsilon^k) = w_i(E) for the trivial line bundle \epsilon.^[11] The cohomology ring of the classifying space BO for real vector bundles is the polynomial algebra H^*(BO; \mathbb{Z}/2) = \mathbb{Z}/2[[w_1, w_2, \dots ]], generated by the universal Stiefel-Whitney classes of the tautological bundle. This formal power series structure arises from the cell decomposition of Grassmannians and the Whitney sum formula, allowing Stiefel-Whitney classes of any bundle to be pulled back from these universal generators via classifying maps.^[11]

Chern classes

Chern classes are characteristic classes defined for complex vector bundles, taking values in the even-degree integer cohomology groups. For a complex vector bundle E of rank n over a smooth manifold M, the total Chern class is c(E) = 1 + c_1(E) + \cdots + c_n(E) \in H^*(M; \mathbb{Z}), where each c_k(E) \in H^{2k}(M; \mathbb{Z}). The first Chern class c_1(E) for a complex line bundle is represented by the curvature form via the formula c_1(E) = \left[ \frac{i}{2\pi} \operatorname{Tr}(F) \right], with F the curvature 2-form of a Hermitian connection on E. In differential geometry, Chern classes are constructed using the Chern-Weil homomorphism for principal U(n)-bundles underlying Hermitian complex vector bundles. Given a connection with curvature form F \in \Omega^2(M, \mathfrak{u}(n)), the total Chern class is represented by the closed (0,2n)-form \det\left(I + \frac{i}{2\pi} F\right) = \sum_{k=0}^n c_k(E), where the individual c_k(E) are the cohomology classes of the corresponding components; this representation is independent of the choice of Hermitian metric and connection. For higher k, c_k(E) is the k-th elementary symmetric polynomial in the eigenvalues of \frac{i}{2\pi} F. Axiomatic characterizations of Chern classes emphasize their uniqueness via naturality and multiplicativity. Specifically, Chern classes satisfy c(E \oplus F) = c(E) \cup c(F) for Whitney sum and are natural under bundle maps, with c(\epsilon^r) = 1 for the trivial rank-r bundle; these axioms, together with normalization on universal bundles, uniquely determine them in integer cohomology. The splitting principle states that any complex vector bundle E pulls back to a sum of line bundles L_1 \oplus \cdots \oplus L_n over some flag manifold cover, so c(E) = \prod_{i=1}^n (1 + c_1(L_i)), reducing computations of symmetric functions in Chern classes to those on line bundles. The Chern classes relate to the Todd class in the Hirzebruch-Riemann-Roch theorem through the Chern character \operatorname{ch}(E) = \sum_{k=0}^\infty \frac{1}{k!} \left( \frac{i}{2\pi} F \right)^k in de Rham cohomology, or topologically as \operatorname{ch}(E) = r + c_1 + \frac{c_1^2 - 2c_2}{2!} + \cdots \in H^*(M; \mathbb{Q}), where r = \operatorname{rank}(E). Hirzebruch's theorem asserts that for a holomorphic vector bundle E on a compact complex manifold M, the holomorphic Euler characteristic satisfies \chi(M, E) = \int_M \operatorname{ch}(E) \cdot \operatorname{Td}(TM), with the Todd class \operatorname{Td}(TM) expressed as a power series in the Chern classes of the tangent bundle. A representative example is the tautological line bundle \mathcal{O}(1) over complex projective space \mathbb{CP}^n, whose first Chern class c_1(\mathcal{O}(1)) generates H^2(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z} as the positive generator corresponding to the Kähler form.

Pontryagin classes

Pontryagin classes are characteristic classes defined for oriented real vector bundles, constructed via the Chern classes of their complexifications. For an oriented real vector bundle E \to M of rank n, the complexification E \otimes \mathbb{C} is a complex vector bundle of the same rank, and the k-th Pontryagin class is

p_k(E) = (-1)^k c_{2k}(E \otimes \mathbb{C}) \in H^{4k}(M; \mathbb{Z}).

The total Pontryagin class is p(E) = 1 + p_1(E) + p_2(E) + \cdots.^[14] These classes exhibit multiplicativity under the Whitney sum: for oriented real vector bundles E and F over the same base, p(E \oplus F) = p(E) \cup p(F). They are also stable, remaining unchanged when E is tensored with a trivial real line bundle, and all classes reside in cohomology degrees that are multiples of 4. For an oriented bundle of even rank $2n, the top Pontryagin class relates to the Euler class by p_n(E) = e(E)^2.^[14]^[15] The Pontryagin classes play a central role in the Hirzebruch signature theorem, which equates the signature of a compact oriented smooth $4k-manifold M to the evaluation of its L-genus on the fundamental class: \sigma(M) = \int_M L(TM), where the L-genus is the multiplicative genus given by formal power series in the Pontryagin classes L(p_1, \dots, p_k), explicitly

L(t) = \frac{\sqrt{t}}{\tanh \sqrt{t}} = 1 + \frac{1}{3}t + \frac{7}{45}t^2 + \cdots

applied fiberwise to the universal bundle. For instance, in dimension 4, this simplifies to \sigma(M) = \frac{1}{3} p_1(TM)[M], linking the first Pontryagin number directly to the signature.^[16] The rational cohomology ring of the classifying space BSO for the special orthogonal group is a polynomial algebra freely generated by the universal Pontryagin classes p_1, p_2, \dots with no torsion elements. More precisely, H^*(BSO; \mathbb{Q}) \cong \mathbb{Q}[p_1, p_2, \dots ].^[15]

Applications

Characteristic numbers

Characteristic numbers are topological invariants obtained by evaluating monomials in characteristic classes on the fundamental class of a closed manifold. For a closed oriented n-dimensional manifold M equipped with a vector bundle E \to M, a characteristic number associated to a monomial \prod c_{i_j}(E) in the characteristic classes of E is given by the pairing \langle \prod c_{i_j}(E), [M] \rangle, where [M] \in H_n(M; \mathbb{Z}) is the fundamental homology class of M, or equivalently the integral \int_M \prod c_{i_j}(E) when the classes are represented by closed differential forms.^[17] Prominent examples include Stiefel-Whitney numbers, which are defined modulo 2 for unoriented manifolds using products of Stiefel-Whitney classes w_i(\tau M) \in H^i(M; \mathbb{Z}/2\mathbb{Z}) paired with the mod 2 fundamental class [M] \in H_n(M; \mathbb{Z}/2\mathbb{Z}), such as \langle w_1 w_{n-1}, [M] \rangle. Chern numbers arise from products of Chern classes c_i(E) \in H^{2i}(M; \mathbb{Z}); for instance, for a complex n-manifold M, the Euler characteristic \chi(M) equals the top Chern number \int_M c_n(TM), where TM is the holomorphic tangent bundle. Pontryagin numbers, defined for oriented $4m-manifolds, involve products of Pontryagin classes p_i(\tau M) \in H^{4i}(M; \mathbb{Z})paired with[M], like the signature \sigma(M) = \langle L_m, [M] \rangle, where L_mis the m-th Hirzebruch L-class, a polynomial in the Pontryagin classesp_1(\tau M), \dots, p_m(\tau M)with rational coefficients (for example, in dimension 4,L_1 = p_1/3$).^[17]^[16]^[17] These numbers exhibit multiplicativity under products of manifolds: for closed oriented manifolds M and N with bundles E \to M and F \to N, the characteristic number of the pullback bundle over M \times N satisfies \int_{M \times N} \prod c_{i_j}(E \oplus F) = \left( \int_M \prod c_{i_j}(E) \right) \left( \int_N \prod c_{k_l}(F) \right), following from the multiplicativity of characteristic classes and the Künneth theorem in cohomology.^[17] Computations of characteristic numbers are facilitated by the Atiyah-Singer index theorem, which equates certain characteristic numbers of the tangent bundle to the analytic index of associated elliptic differential operators on the manifold, thereby providing a bridge between topology and analysis.^[18] Vanishing theorems highlight constraints on these invariants; for example, the Euler characteristic \chi(M) vanishes for any closed odd-dimensional manifold M, as \chi(M) = 0 in odd dimensions.^[17]^[19] These characteristic numbers also determine the cobordism class of the manifold in the corresponding bordism groups; for example, the unoriented bordism ring is isomorphic to the polynomial ring on the Stiefel-Whitney classes of the Stiefel manifolds.^[17]

Topological invariants and obstructions

Characteristic classes play a central role in algebraic topology by providing obstructions to the existence of certain geometric structures on manifolds and bundles, such as sections, reductions of structure groups, and specific types of maps. For oriented real vector bundles, the Euler class e(E) serves as the primary obstruction to the existence of a nowhere-vanishing section; specifically, a continuous nowhere-zero section exists if and only if e(E) = 0 in the cohomology group H^n(B; \mathbb{Z}), where n is the rank of the bundle and B is the base space.^[2] This obstruction arises from the fact that the Euler class is the image under the connecting homomorphism in the long exact sequence of the pair (DE, SE), where DE and SE are the disk and sphere bundles associated to E, capturing the failure of a section to extend over the entire space.^[20] In the classification of vector bundles, characteristic classes detect possible reductions of the structure group. For real vector bundles with structure group O(n), the vanishing of the second Stiefel-Whitney class w_2(E) \in H^2(B; \mathbb{Z}/2) is the necessary and sufficient condition for reducing the structure group to SO(n), thereby endowing the bundle with an orientation.^[21] More generally, the existence of a spin structure on a Riemannian manifold M, which reduces the structure group of the tangent bundle from SO(n) to Spin(n), is obstructed precisely by the vanishing of w_2(TM); if w_2(TM) = 0, then such a reduction exists, allowing the definition of spinors and Dirac operators on M.^[21] These obstructions extend to higher structure groups, where higher Stiefel-Whitney classes w_k(E) for k \geq 3 provide further barriers to reductions like special orthogonal or spin structures. For manifolds, characteristic classes yield powerful invariants that classify topological properties. The first Stiefel-Whitney class w_1(TM) of the tangent bundle determines orientability: a manifold M is orientable if and only if w_1(TM) = 0 in H^1(M; \mathbb{Z}/2), as this class measures the obstruction to a consistent choice of orientation across the manifold.^[22] In the complex setting, Chern classes classify almost complex and complex structures; for a smooth manifold to admit an almost complex structure, the Stiefel-Whitney classes must satisfy certain integrality conditions related to the Chern classes of the associated complex bundle, and the full Chern classes c_k(TM \otimes \mathbb{C}) distinguish non-isomorphic complex structures up to diffeomorphism in low dimensions.^[23] Characteristic classes also feature prominently in embedding and immersion theory. In the Hirsch-Smale theory, which classifies immersions of manifolds via homotopy classes of bundle monomorphisms, the Stiefel-Whitney classes of the virtual normal bundle provide obstructions to the existence of immersions; for instance, an immersion of an m-manifold into \mathbb{R}^n with m < n exists if the stable normal bundle's Stiefel-Whitney classes vanish in degrees greater than n - m, reducing the problem to algebraic topology via the Smale-Hirsch theorem.^[24] This framework implies that closed manifolds immerse in Euclidean space provided their dimensions satisfy Whitney's inequalities, with SW classes quantifying the topological barriers. Finally, in gauge theory, characteristic classes classify moduli spaces of connections and instantons. In Donaldson theory for four-manifolds, anti-self-dual instantons on principal SU(2)-bundles are classified by the second Chern number c_2(E), which serves as the instanton number and determines the dimension of the moduli space; the Donaldson invariants, derived from these moduli spaces, detect smooth structures via integrals over instanton contributions weighted by Chern classes.^[25] These invariants, which refine earlier characteristic numbers like the signature, provide obstructions to exotic smooth structures on manifolds such as \mathbb{CP}^2 \# k \overline{\mathbb{CP}^2}.^[25]

References

[1]
[PDF] Characteristic classes - Arun Debray
Characteristic classes are natural cohomology classes of vector bundles. Let's exposit this a bit. Definition 1.1. Recall that a (real) vector bundle over a ...
[2]
[PDF] Characteristic Classes, Principal Bundles, and Curvature
Jul 26, 2024 · These are the lecture notes for my 2024 Summer Minicourse on Character- istic classes, taking place at UT Austin. Section 2.1 is completely ...
[3]
[PDF] Lecture 7: Characteristic classes - Harvard Mathematics Department
Sep 25, 2012 · In this lecture we describe some basic techniques in the theory of characteristic classes, mostly focusing on Chern classes of complex ...
[4]
[PDF] CHARACTERISTIC CLASSES
The theory of characteristic classes began in the year 1935 with almost simultaneous work by HASSLER WHITNEY in the United States and. EDUARD STIEFEL in ...
[5]
[PDF] Algebraic Topology - Cornell Mathematics
This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in ...Missing: primary | Show results with:primary
[6]
Shiing-shen Chern - Biography - MacTutor - University of St Andrews
Shiing-shen Chern was a Chinese mathematician who made important contributions to geometry and algebraic topology. Thumbnail of Shiing-shen Chern View twelve ...<|control11|><|separator|>
[7]
[PDF] Types and Properties of Characteristic Classes
Sep 15, 2017 · Pontryagin's work goes back to (1942)[2]. In (1946), Chern defined characteristic classes for complex vector bundles, and showed that complex ...
[8]
ARITHMETIC GENERA AND THE THEOREM OF RIEMANN-ROCH ...
ARITHMETIC GENERA AND THE THEOREM OF RIEMANN-ROCH FOR ALGEBRAIC VARIETIES. Friedrich HirzebruchAuthors Info & Affiliations. February 15, 1954.
[9]
None
### Summary of Introduction and Definition of Characteristic Classes
[10]
[PDF] K-theory and Characteristic Classes: A homotopical perspective
gives its algebraic definition. Section 9.2 discusses the relationship of topo- logical K-theory and characteristic classes, defines the Chern character, and.
[11]
[PDF] Characteristic Classes
Sep 28, 2021 · 3It is customary in algebraic topology to call this the “weak topology,” a weak topology ... space Grn(R∞) has the direct limit topology by ...
[12]
CHARACTERISTIC CLASSES - Project Euclid
Dold also has results on characteristic classes in generalized cohomology theories. The classical techniques used may be found in Milnor's notes [29] ...
[13]
[PDF] User's Guide to Spectral Sequences - UiO
Jul 17, 2000 · ... spectral sequences, as well as the important class of nilpotent spaces. ... transgression. 185. 6.3. Classifying spaces and characteristic ...
[14]
[PDF] Version 2.2, November 2017 Allen Hatcher Copyright c 2003 by ...
Besides tensor product of vector bundles, another construction useful in K–theory ... Characteristic classes are cohomology classes associated to vector bundles.
[15]
[PDF] An Introduction to Characteristic Classes - Derek Sorensen
Dec 18, 2017 · In particular we are interested in the rational cohomology ring H∗(BSO(n);Q), as SO(n) is the structure group of such vector bundles. In a ...
[16]
[PDF] The Hirzebruch Signature Formula (Lecture 25)
Mar 30, 2011 · Taking the limit as n → ∞, we obtain a formula for l(ζ) in terms of the Pontryagin classes of ζ (which is valid for any orientable real vector ...
[17]
[PDF] Characteristic classes Robert R. Bruner Michael Catanzaro J. Peter ...
We develop the classical theory of characteristic classes. Our procedure is simultaneously to compute the cohomology of the relevant classifying spaces and to ...
[18]
[PDF] The Atiyah-Singer Index Theorem*
Here is a brief outline to the paper. In Section 1, we review some basic facts concerning. Clifford algebras and spin structures.
[19]
[1302.5588] Euler Characteristic in Odd Dimensions - arXiv
Feb 22, 2013 · It is well known that the Euler characteristic of an odd dimensional compact manifold is zero. An Euler complex is a combinatorial analogue of a compact ...
[20]
[PDF] characteristic classes and obstruction theory - UChicago Math
This paper proves the following obstruction property for Stiefel-Whitney classes: if wiξ = 0 for an n-dimensional bundle ξ, then there cannot exist n − i + 1 ...Missing: immersions | Show results with:immersions
[21]
[PDF] Spin Structures and the Second Stiefel-Whitney Class - UTK Math
With this definition of w1, clearly it vanishes exactly if the structure group of. PO(E) can be reduced to SOn, that is, exactly if E is orientable. This ...
[22]
[PDF] Yikai Teng - A note on Stiefel-Whitney Classes
This is a lecture note of the first talk of the seminar “Characteristic Classes”, and we will focus on the definition (axioms) of Stiefel-Whitney Classes along ...<|control11|><|separator|>
[23]
[PDF] Chern Classes of a Complex Vector Bundle
The proof and the computations are based on the splitting principle, which, roughly speaking, states that if a polynomial identity in the Chern classes holds ...Missing: source | Show results with:source
[24]
[PDF] Immersions of Manifolds - Morris W. Hirsch
Aug 9, 2000 · The Whitney-. Graustein theorem [13] classifies immersions of the circle S¹ in the plane E up to regular homotopy, which is a homotopy f, with ...
[25]
[PDF] An introduction to Donaldson-Witten theory
Aug 22, 2013 · An important set of topological invariants of X is given by the characteristic classes of its real tangent bundle. The most elementary ones ...