Cobordism
In mathematics, particularly algebraic topology, cobordism is an equivalence relation defined on the class of compact smooth manifolds of a fixed dimension n, where two such manifolds M and N (possibly with additional structure, such as orientation or a map to a space X) are cobordant if their disjoint union bounds a compact smooth (n+1)-manifold W, meaning \partial W = -M \sqcup N (with the negative sign indicating reversed orientation if applicable).[1] This relation partitions the manifolds into equivalence classes that form abelian groups under disjoint union, known as cobordism groups \mathfrak{N}_n(X) or Q_n for oriented cases, providing a framework to classify manifolds up to "bounding" behavior.[2] The theory was pioneered by René Thom in his 1954 paper Quelques propriétés globales des variétés différentiables, where he established that these groups are isomorphic to the stable homotopy groups of Thom spaces associated to the universal bundle over the classifying spaces BO(k) or BSO(k), linking cobordism directly to homotopy theory.[1][3] Thom's foundational work distinguished between unoriented (mod 2) cobordism and oriented cobordism, computing the former using Stiefel-Whitney numbers (a manifold bounds iff all are zero): the unoriented cobordism ring is a polynomial algebra over \mathbb{Z}/2 on generators in degrees not of the form $2^k - 1, while oriented groups are more complex but finitely generated in low dimensions.[2] Subsequent developments extended cobordism to structured variants, such as complex cobordism (MU), introduced by John Milnor in 1959 and revolutionized by Daniel Quillen's 1969 theorem connecting it to the universal formal group law, which underpins its role as the "universal" generalized cohomology theory.[4] These extensions, including Spin-cobordism and String-cobordism, incorporate additional bundle structures and have been computed using Adams spectral sequences adapted to cobordism contexts, as in Sergei Novikov's 1967 work. Cobordism theory has profoundly influenced algebraic topology and beyond, enabling the classification of manifolds via geometric invariants, underpinning surgery theory for distinguishing homotopy equivalent manifolds, and contributing to the Atiyah-Singer index theorem through Thom's transversality techniques.[3] It also intersects with physics via topological quantum field theories (TQFTs), where the cobordism hypothesis formalizes extended TQFTs as representations of cobordism categories, and with algebraic geometry through connections to formal groups and K-theory.[5] Thom's innovations earned him the 1958 Fields Medal, highlighting cobordism's role in reshaping global manifold properties and generalized homology.[3]Basic Concepts
Manifolds
A smooth n-manifold is a second-countable Hausdorff topological space M that is locally Euclidean of dimension n, meaning every point in M has a neighborhood homeomorphic to an open subset of \mathbb{R}^n, equipped with a maximal smooth atlas. An atlas consists of charts (U_\alpha, \phi_\alpha) where each \phi_\alpha: U_\alpha \to \mathbb{R}^n is a homeomorphism to an open set, and transition maps \phi_\beta \circ \phi_\alpha^{-1} between overlapping charts are smooth (C^\infty) diffeomorphisms. The maximal atlas is the unique largest compatible collection containing any given smooth atlas, ensuring the smooth structure is well-defined independent of chart choices.[6][7] Manifolds are distinguished by compactness: a manifold is compact if it is compact as a topological space, which implies it is closed and bounded in any embedding into Euclidean space. Non-compact manifolds, such as \mathbb{R}^n itself, extend infinitely and lack this boundedness. In cobordism theory, compact manifolds are emphasized because cobordisms relate compact objects, allowing for controlled geometric relations without infinite extent.[8][9] An orientation on a smooth n-manifold M is a consistent choice of ordered basis for each tangent space T_pM at points p \in M, such that the change of basis between nearby points has positive determinant, or equivalently, a maximal atlas where all transition maps are orientation-preserving (determinant >0). This global consistency distinguishes orientable manifolds, like the sphere S^n for all n, from non-orientable ones, such as the real projective plane \mathbb{RP}^2.[10][11] The dimension n of a manifold is fixed by its local Euclidean model, and manifolds may have boundaries: a smooth manifold with boundary admits charts mapping to open subsets of the half-space \mathbb{R}^n_+ = \{(x_1, \dots, x_n) \in \mathbb{R}^n \mid x_n \geq 0\}, with the boundary \partial M consisting of points mapping to the hyperplane x_n = 0. Closed manifolds, or boundaryless manifolds, have empty boundary and are the primary objects in many studies; representative examples include the n-sphere S^n = \{x \in \mathbb{R}^{n+1} \mid \|x\| = 1\}, the n-torus T^n = S^1 \times \cdots \times S^1 (n factors), and the real projective space \mathbb{RP}^n, all of which are compact closed smooth manifolds of dimension n.[12][7][13]Cobordisms
In differential topology, an n-cobordism between two closed oriented (n-1)-manifolds M and N is defined as a compact oriented n-manifold W equipped with an orientation-preserving diffeomorphism from its boundary \partial W to the disjoint union -M \sqcup N, where -M denotes M with its orientation reversed.[14] This construction captures the idea that M and N serve as the "boundaries" of a higher-dimensional manifold W, generalizing the notion of manifolds bounding each other in a relational sense. The boundary operator satisfies the equation \partial W = M \sqcup (-N) under the convention that the incoming boundary component M receives the reversed orientation while the outgoing N preserves it, ensuring consistency in orientation conventions across gluings.[14] Two closed oriented (n-1)-manifolds M and N are said to be cobordant, denoted M \sim N, if there exists an n-cobordism W between them. This relation defines an equivalence relation on the set of closed oriented (n-1)-manifolds: it is reflexive, as the trivial cobordism M \times [0,1] provides a cylinder connecting M to itself; symmetric, since reversing the orientation on W yields a cobordism from N to M; and transitive, by gluing cobordisms end-to-end along common boundary components to form a new cobordism.[14] Product cobordisms like M \times [0,1] serve as basic examples of trivial cases, illustrating how the relation extends the diffeomorphism equivalence to a coarser topological structure. The cobordism relation respects disjoint unions, meaning if M \sim M' and N \sim N', then M \sqcup N \sim M' \sqcup N'.[14] Consequently, the set of cobordism equivalence classes of closed oriented n-manifolds inherits a commutative monoid structure under the operation induced by disjoint union, with the empty manifold serving as the identity element. This monoidal structure underpins the algebraic framework of cobordism theory, allowing equivalence classes to be combined additively while preserving the bounding relations.[14]Examples
A fundamental example of a cobordism arises with odd-dimensional spheres. For any integer k \geq 0, the sphere S^{2k+1} is cobordant to the empty manifold via the (2k+2)-dimensional ball D^{2k+2}, whose boundary is precisely S^{2k+1}. This illustrates how the sphere serves as the boundary of a higher-dimensional disk, establishing the triviality of odd-dimensional classes in oriented cobordism.[15] In low dimensions, simple disk-like cobordisms provide intuition. The 0-dimensional sphere S^0, consisting of two points, is cobordant to the empty set through the 1-dimensional disk (an interval), where the two endpoints form S^0. In dimension 2, the real projective plane \mathbb{RP}^2 does not bound a compact 3-manifold on its own, reflecting its nontrivial class in unoriented cobordism; however, the connected sum of two copies of \mathbb{RP}^2 does bound a compact 3-manifold, such as the twisted I-bundle over the Klein bottle.[15][16] A classic pair of non-cobordant manifolds in unoriented 2-dimensional cobordism consists of the 2-sphere S^2 and the real projective plane \mathbb{RP}^2. While S^2 bounds the 3-ball, \mathbb{RP}^2 cannot bound any compact 3-manifold, as their classes differ in the cobordism group \mathbb{Z}/2\mathbb{Z}.[15] Visualizations in low dimensions often depict cobordisms as familiar shapes. For instance, a "cap" illustrates a manifold bounding the empty set, such as the disk capping a circle S^1. In contrast, "pants" represent a cobordism between two circles and one circle, formed by a pair-of-pants surface connecting the incoming boundaries to the outgoing one. These diagrams emphasize the relational aspect of cobordism without requiring embeddings in higher space.[15] Basic invariants like the Euler characteristic modulo 2 distinguish non-cobordant manifolds in unoriented cobordism. For example, S^2 has Euler characteristic \chi(S^2) = 2 \equiv 0 \pmod{2}, while \mathbb{RP}^2 has \chi(\mathbb{RP}^2) = 1 \equiv 1 \pmod{2}, confirming they are not cobordant; the connected sum of two \mathbb{RP}^2 has \chi = 0 \pmod{2} and thus bounds. This mod-2 Euler characteristic serves as a complete invariant for dimension 2 unoriented cobordism.[17][15]Formal Framework
Terminology
In cobordism theory, bordism refers to the one-sided relation in which a closed manifold bounds a higher-dimensional compact manifold, meaning it is the boundary of some manifold without an additional pairing component. In contrast, cobordism denotes the two-sided equivalence relation between two closed manifolds M and N, where M is cobordant to N if there exists a compact manifold W such that the boundary \partial W is the disjoint union of M and N (in the unoriented case) or M \sqcup (-N) (in the oriented case, with -N denoting N with reversed orientation). This equivalence is reflexive, symmetric, and transitive, forming the basis for cobordism groups.[18][19] Closed cobordisms restrict attention to compact manifolds without boundary as the objects, connected via compact cobordisms whose boundaries are the disjoint unions of these closed manifolds; this setup emphasizes equivalence classes of closed manifolds. Open cobordisms, by comparison, extend the framework to include manifolds with boundary or non-compact examples, allowing for more general relations but often requiring additional structure like asymptotic behavior at infinity.[19][20] The dimension convention specifies that an n-cobordism is an n-dimensional compact manifold whose boundary decomposes into components that are (n-1)-dimensional manifolds, establishing the cobordism relation between these lower-dimensional objects.[18] Standard notation for unoriented cobordism uses \Omega_n to denote the cobordism group in dimension n, comprising equivalence classes of n-dimensional closed unoriented manifolds under the cobordism relation, with group operation given by disjoint union. The reduced cobordism group \tilde{\Omega}_n (often for a pointed space) is the kernel of the augmentation map \Omega_n \to \Omega_0, excluding the trivial class.[18] The Pontryagin-Thom construction offers a dual perspective, realizing cobordism classes geometrically as homotopy classes of maps into Thom spaces associated with stable normal bundles, thereby connecting cobordism to stable homotopy theory.[18][19]Cobordism Groups
The cobordism groups provide a formal algebraic structure for classifying closed manifolds up to the cobordism relation. For each nonnegative integer n, the group \Omega_n consists of equivalence classes of closed n-dimensional manifolds, where two such manifolds M and N represent the same class if they bound a common compact (n+1)-dimensional manifold W, meaning \partial W = M \sqcup N. This construction captures the intuitive notion that manifolds are "equivalent" if they can be connected through an intermediate manifold without boundary issues, forming the foundational objects in cobordism theory. The group operation on \Omega_n is induced by the disjoint union of manifolds: for classes [M] and [N] in \Omega_n, their sum is defined by [M] + [N] = [M \sqcup N], where M \sqcup N denotes the disjoint union, which is itself a closed n-manifold. Since the theory is unoriented, the groups are 2-torsion: [M] + [M] = [M \sqcup M] = 0, as M \sqcup M bounds a compact (n+1)-manifold (the unoriented double of M). Thus, the additive inverse satisfies -[M] = [M]. The zero element is the equivalence class of the empty manifold, which bounds the empty (n+1)-manifold. This structure makes \Omega_n an abelian group, as disjoint union commutes up to diffeomorphism: M \sqcup N is diffeomorphic to N \sqcup M, ensuring commutativity. These operations are well-defined on equivalence classes because cobordisms respect disjoint unions. Cobordism classes in \Omega_n are invariant under diffeomorphisms of manifolds, meaning that if M and M' are diffeomorphic, then [M] = [M']; this extends to homotopies, as diffeomorphisms can be isotoped through homotopies in the appropriate settings without altering the cobordism relation. A key property established in the theory is the finiteness of these groups: Thom proved that each \Omega_n is finitely generated as an abelian group, implying that the classification of n-manifolds up to cobordism reduces to a finite set of generators and relations, which has profound implications for computations in algebraic topology.Variants
Cobordism theory extends beyond the classical smooth category to other geometric settings, including piecewise linear (PL) and topological (TOP) manifolds, as well as variants incorporating additional bundle structures. In the smooth category, manifolds are equipped with C∞ atlases, while PL cobordism uses simplicial complexes with linear simplices, and topological cobordism relies on homeomorphisms without additional structure. These categories differ significantly in low dimensions: for instance, in dimension 4, topological manifolds admit exotic smooth structures that are not smoothly isotopic, leading to distinct cobordism relations.[21] However, by the h-cobordism theorem and smoothing theory, the PL and smooth categories coincide for dimensions n ≥ 5, and topological cobordism aligns with them in simply connected cases above dimension 5.[22] Framed cobordism considers n-manifolds embedded in Euclidean space ℝ^{n+k} with a trivialization of the normal bundle, equivalent to a stable trivialization of the tangent bundle.[23] Two such framed manifolds are cobordant if they bound a compact (n+1)-manifold with a compatible framing on its normal bundle. The framed cobordism groups Ω_n^{fr} are isomorphic to the stable homotopy groups of spheres π_n^s via the Pontryagin-Thom construction, which maps framed manifolds to homotopy classes in Thom spaces.[23] This isomorphism highlights framed cobordism's role in computing stable homotopy, as established by Pontryagin in his duality theorem.[24] Spin cobordism restricts to spin manifolds, which are oriented Riemannian manifolds whose tangent bundle admits a spin structure—a lift of the structure group from SO(n) to Spin(n), existing precisely when the second Stiefel-Whitney class w_2 vanishes.[25] Two closed spin manifolds are spin cobordant if their disjoint union bounds a compact spin manifold preserving the spin structures. The spin cobordism groups Ω_n^{Spin} are computed via Thom spectra and relate to index theory: for example, the Â-genus of a 4k-dimensional spin manifold equals the index of the Dirac operator, as per the Atiyah-Singer index theorem.[25][26] The cobordism groups differ across categories in low dimensions but stabilize in higher ones; specifically, the topological unoriented cobordism groups Ω_n^T are isomorphic to the smooth groups Ω_n for n ≥ 5, while discrepancies arise in dimensions 1 through 4 due to the absence of smoothing in topological settings.[19] For instance, in dimension 4, topological cobordism allows more classes not representable by smooth manifolds.[21] Rational cobordism arises by tensoring the complex cobordism ring MU_* with ℚ, simplifying its torsion-free structure to a polynomial algebra over ℚ generated by the classes of complex projective spaces [ℂℙ^n] in degrees 2n.[27] This rationalization, due to Quillen's computation of MU_* via formal group laws, reveals that rational cobordism classes are determined by Chern numbers without torsion obstructions.[28]Constructions
Surgery
Surgery is a fundamental technique in differential topology for modifying smooth manifolds to construct cobordisms and study their classification up to cobordism. In an n-dimensional manifold M^n, surgery along an embedded k-sphere proceeds by selecting a smoothly embedded sphere S^k with a framing of its normal bundle, which provides a tubular neighborhood diffeomorphic to S^k \times D^{n-k}. The interior of this neighborhood is excised, and a handle D^{k+1} \times S^{n-k-1} is glued along the boundary S^k \times S^{n-k-1} using the framing to match the orientations and bundle structures. This yields a new manifold M' of the same dimension.[29][30] The trace of the surgery—the original manifold with the attached handle—forms a cobordism W from M to M', where \partial W = -M \sqcup M' (with the negative orientation on M). This construction preserves the homotopy type of the manifold in dimensions greater than or equal to 5, allowing surgeries to systematically kill homotopy groups below the middle dimension and relate manifolds in the same cobordism class. The steps involve: (1) embedding the framed sphere S^k into M, ensuring the normal bundle admits the required framing; (2) excising the open tubular neighborhood; and (3) attaching the handle via a diffeomorphism of the boundaries. In high dimensions (n ≥ 5), such surgeries can be performed without altering the diffeomorphism type beyond the intended modification, facilitating the classification of manifolds up to h-cobordism.[29][31] The Kervaire invariant plays a crucial role as a surgery obstruction in distinguishing cobordism classes, particularly for framed manifolds in dimensions where the stable homotopy groups of spheres exhibit 2-torsion, such as dimensions 3 and 7 mod 8. Defined as the Arf invariant of a quadratic form \phi: H_k(M; \mathbb{Z}/2) \to \mathbb{Z}/2 associated to the intersection pairing on the homology, it is given by \kappa(M) = \sum \phi(a_i) \phi(b_i) \pmod{2}, where \{a_i, b_i\} is a symplectic basis for the hyperbolic plane decomposition. A nonzero Kervaire invariant obstructs the existence of a framing reversal or certain handle attachments, preventing the manifold from being cobordant to the standard sphere in those classes; for example, it detects the nontrivial element in the 10-dimensional homotopy sphere group \Theta_{10} \cong \mathbb{Z}/6.[29][30] Surgery also impacts the fundamental group, computable via the Seifert-van Kampen theorem. For k=1 surgery on an embedded circle representing an element \gamma \in \pi_1(M), the fundamental group of the resulting manifold M' is the quotient \pi_1(M') \cong \pi_1(M) / \langle \langle i_*(\gamma) \rangle \rangle, where i_* is the inclusion-induced map and \langle \langle \cdot \rangle \rangle denotes the normal subgroup generated by the image; this kills the subgroup generated by \gamma if the embedding is nullhomotopic in the complement. In higher k, the effect is trivial on \pi_1 if k ≥ 2, as the attaching sphere does not intersect the 1-skeleton.[32][29]Morse Functions
A Morse function on a manifold M is a smooth map f: M \to \mathbb{R} such that all its critical points are non-degenerate, meaning that at each critical point p, the Hessian matrix \left( \frac{\partial^2 f}{\partial x_i \partial x_j}(p) \right) has non-zero determinant. Critical points occur where the differential satisfies df_p = 0, or equivalently, where the gradient \nabla f vanishes. In the context of cobordisms, consider a compact smooth manifold with boundary W^{n+1} whose boundary components are V_0 \sqcup (-V_1), forming a cobordism from V_0 to V_1. A Morse function on the triad (W; V_0, V_1) is a smooth map f: W \to [a, b] such that f^{-1}(a) = V_0, f^{-1}(b) = V_1, the restriction of f to each boundary component is a submersion, and all critical points lie in the interior of W with non-degenerate Hessians. The index \lambda(p) of a critical point p is defined as the number of negative eigenvalues of the Hessian at p, which by the Morse lemma locally coordinates f near p as f(x) = f(p) - \sum_{i=1}^{\lambda} x_i^2 + \sum_{i=\lambda+1}^{n+1} x_i^2. This local form highlights the quadratic nature of the function around critical points, ensuring transverse level sets away from them. The gradient flow of f, generated by a gradient-like vector field \xi satisfying \xi(f) > 0 outside critical points, governs the evolution of level sets f^{-1}(c). As the parameter c increases through a critical value, the topology of the sublevel set f^{-1}((-\infty, c]) changes by attaching a handle of index \lambda, effectively realizing the cobordism through a sequence of such attachments along the level sets. Any cobordism admits a Morse function whose critical points encode births and deaths—pairwise creations or annihilations of critical points of consecutive indices in generic perturbations—corresponding to the analytic realization of the cobordism's structure.[33] The indices \lambda of these critical points satisfy the Morse inequalities, which relate the number m_k of critical points of index k to the Betti numbers b_k of W: specifically, the weak inequalities m_k \geq b_k and the strong inequalities \sum_{k=0}^j (-1)^{j-k} m_k \geq \sum_{k=0}^j (-1)^{j-k} b_k for all j. This analytic approach via Morse functions serves as a continuous counterpart to discrete constructions in cobordism theory.Handlebodies and Geometry
In the context of cobordism theory, a handlebody decomposition offers a structured geometric representation of a cobordism W between two compact manifolds with boundary, typically expressed relative to one boundary component, say the incoming boundary \partial_0 W. This decomposition begins with the trivial cobordism \partial_0 W \times [0,1] and proceeds by successively attaching handles, where an n-dimensional k-handle is the product D^k \times D^{n-k}, attached along its boundary component S^{k-1} \times D^{n-k} via an embedding into the current boundary of the partially constructed manifold. Handles are attached up to index at most n/2, as higher-index handles can be dualized to lower ones via the cocore structure, ensuring a minimal and symmetric presentation.[34] The attachment of handles in a cobordism is intimately linked to Morse theory, where a Morse function on W relative to \partial_0 W—with non-degenerate critical points—induces the decomposition: a minimum (index 0 critical point) corresponds to attaching a 0-handle (a disk D^n), while a saddle point of index k (1 ≤ k ≤ n-1) attaches a k-handle, thickening the level sets across the critical value. This process builds the cobordism incrementally, with sublevel sets W^c for regular values c forming the stages between attachments, and the outgoing boundary \partial_1 W emerging as the top level set. Such decompositions exist for any smooth cobordism, providing a CW-complex structure homotopy equivalent to W.[35] Geometrically, cobordisms admit simplifications through handle slides and cancellations, which preserve the diffeomorphism type while refining the decomposition. A handle slide involves isotoping the attaching sphere of one handle over the belt sphere of another, effectively changing the attachment without altering the manifold; for instance, sliding a k-handle over a j-handle (j < k) adjusts the framing and connectivity. Cancellations occur when a k-handle and (k+1)-handle pair intersect transversely at a single point between their attaching and belt spheres, allowing their removal via a diffeomorphism to the pre-attachment manifold. These operations enable the reduction of redundant handles, yielding a streamlined geometric model of the cobordism.[36] In four dimensions, these geometric manipulations are formalized by Kirby calculus, a set of moves on handlebody presentations—primarily handle slides and the creation/cancellation of disjoint 1/2-handle pairs—that preserve the diffeomorphism type of the 4-manifold or cobordism. Kirby diagrams, consisting of framed links representing 1- and 2-handles (with 0- and 3-handles implicit), visualize these relations, allowing classification up to diffeomorphism via link isotopies and changes in framing. This calculus is particularly powerful for 4D cobordisms, connecting handle decompositions to link theory and facilitating computations of smooth structures. Visually, a cobordism as a handlebody is depicted with the incoming boundary at the bottom, handles "growing" upward like protrusions or tunnels connecting to the outgoing boundary at the top, illustrating the connectivity and topology transfer between the boundaries through the attached structures.[34]History
Origins
The origins of cobordism theory trace back to the late 19th century, rooted in Henri Poincaré's foundational work on algebraic topology. In his 1895 paper "Analysis Situs," Poincaré introduced concepts central to homology theory, attempting to define homology groups using cycles represented by manifolds rather than simplicial chains, with the key idea that certain manifolds "bound" higher-dimensional ones, forming the kernel of boundary operators in a chain complex. This approach, though ultimately unsuccessful in fully replacing simplicial homology, laid the groundwork for viewing manifolds up to equivalence relations involving bounding structures, influencing later topological invariants. During the 1930s, advances in differential topology further shaped these ideas through work on manifold immersions and the notion of general position, which prefigured transversality. Hassler Whitney's embedding and immersion theorems, particularly his 1936 result showing that any smooth n-manifold embeds in Euclidean space of dimension 2n+1, emphasized the role of generic maps and intersections between manifolds, providing tools to study how lower-dimensional manifolds could be realized without unintended singularities. These developments shifted focus toward the geometric relations between manifolds, setting the stage for equivalence classes based on bounding behaviors rather than isolated embeddings. The post-World War II era marked a significant expansion in algebraic topology, driven by the growth of homotopy theory and the influx of mathematicians into the field. From the mid-1940s onward, efforts by figures like Samuel Eilenberg and Norman Steenrod formalized homotopy groups and spectral sequences, creating a richer framework for classifying spaces and maps that highlighted the limitations of classical homology in distinguishing smooth structures on manifolds.[37] This boom, fueled by increased academic resources and international collaboration, encouraged explorations into finer invariants beyond homotopy, particularly for smooth manifolds. A pivotal moment came in 1956 with John Milnor's discovery of exotic spheres, smooth manifolds homeomorphic but not diffeomorphic to the standard 7-sphere, constructed via the total space of certain sphere bundles over S^4. This observation revealed that smooth structures on homotopy spheres could vary, prompting intensified study of cobordism as a framework to classify such manifolds up to diffeomorphism via bounding relations, bridging differential geometry and algebraic topology.Key Developments
In 1954, René Thom established the finiteness of the unoriented cobordism groups, proving that the group N_n of n-dimensional unoriented manifolds up to cobordism is finite for each n, with the rank determined by the number of monomials of degree n in the Stiefel-Whitney classes.[38] This result relied on a geometric approach using transversality to count intersections with generic submanifolds, providing a foundational computational tool for cobordism theory. Thom's work culminated in his invited address at the 1958 International Congress of Mathematicians in Edinburgh, where he outlined the implications of cobordism for classifying smooth manifolds and received the Fields Medal for these contributions.[39] Building on Thom's framework, computations of the oriented cobordism ring advanced rapidly in the late 1950s. John Milnor demonstrated in 1959 that the oriented cobordism groups \Omega_n vanish for odd n and computed low-dimensional terms, showing the ring structure begins as a polynomial algebra generated by classes from complex projective spaces. Concurrently, Patrick Conner and Edwin Floyd initiated the study of complex-oriented cobordism, laying groundwork for ring computations that C. T. C. Wall extended by relating oriented and unoriented groups via exact sequences. These efforts established that the oriented cobordism ring \Omega_* is generated by manifolds of dimension $4k, with no torsion in even dimensions. In the 1960s, J. Frank Adams developed the Adams spectral sequence, adapting it to compute cobordism groups through the homotopy groups of Thom spectra, which bridged stable homotopy theory and bordism computations.[40] This tool enabled systematic determination of cobordism via Ext groups in the Steenrod algebra, resolving previously inaccessible higher-dimensional terms and influencing subsequent classifications. A landmark result came in 1963 from Michel Kervaire and John Milnor, who classified exotic spheres by defining the group \Theta_n of h-cobordism classes of homotopy n-spheres and proving it finite for n \geq 5, with \Theta_n fitting into an exact sequence involving the image of the J-homomorphism and Bernoulli numbers.[41] Their analysis revealed 28 distinct smooth structures on the 7-sphere, highlighting the exotic nature of differentiable manifolds beyond topological equivalence. By 1965, Sergei Novikov computed the rational oriented cobordism ring, showing \Omega_* \otimes \mathbb{Q} is a polynomial algebra freely generated by classes in dimensions 4, 8, 12, 16, and so on, using Adams operations on cobordism to detect generators.[42] That same year, Michael Atiyah and Isadore Singer's index theorem for elliptic operators on compact manifolds incorporated cobordism invariance, expressing the analytic index as an integral of characteristic classes over the manifold, thus linking differential operators to bordism groups.[43] These developments coincided with the emergence of surgery theory and Morse functions as complementary tools for manifold classification.Specific Theories
Unoriented Cobordism
Unoriented cobordism provides the simplest case of bordism theory, where orientation is ignored. The unoriented cobordism group in dimension n, denoted \Omega_n^U, is the abelian group generated by isomorphism classes of closed n-dimensional smooth manifolds, with the relation that two manifolds represent the same class if their disjoint union bounds a compact (n+1)-dimensional smooth manifold (without requiring orientation on the bounding manifold). The group operation is induced by disjoint union, and since twice any class is zero (as the double of a manifold bounds its product with an interval), each \Omega_n^U is a vector space over \mathbb{F}_2. There is also a commutative graded ring structure on \Omega_*^U = \bigoplus_n \Omega_n^U, with multiplication given by the Cartesian product of manifolds, which preserves the dimension additively.[44] René Thom's groundbreaking computation in 1954 identified \Omega_n^U with the n-th homotopy group of the Thom spectrum \mathrm{MO}, via the Pontryagin-Thom construction, which equates cobordism classes with stable maps to the Grassmannians (or their one-point compactifications, the Thom spaces). Thom proved that these groups are finite $2-torsion groups by showing that cobordism classes are completely determined by Stiefel-Whitney numbers: for a closed n-manifold M, the Stiefel-Whitney numbers \langle w_{i_1} \cdots w_{i_k} [M] \rangle, where w_j are the Stiefel-Whitney classes of the tangent bundle and the sum of indices is n, take values in \mathbb{Z}/2 and classify [M] up to unoriented cobordism. This yields an isomorphism \Omega_n^U \cong (\mathbb{Z}/2)^{b_n}, where b_n is the number of independent Stiefel-Whitney monomials of degree n, corresponding to the dimension of the space of Ad-invariant polynomials on the Lie algebra of O(\infty) modulo decomposables. The full ring structure of \Omega_*^U is a polynomial algebra over \mathbb{F}_2 generated by one element x_i in each positive degree i that is not of the form $2^k - 1 for integer k \geq 1 (i.e., skipping degrees 1, 3, 7, 15, ...), so \Omega_*^U \cong \mathbb{F}_2[x_2, x_4, x_5, x_6, x_8, x_9, x_{10}, \dots ]. The additive structure in low dimensions follows from the number of monomials of each degree; Thom computed these explicitly up to dimension 8 using Stiefel-Whitney invariants, with later confirmations via the Adams spectral sequence. Generators in these degrees are represented by specific manifolds: for example, x_2 = [\mathbb{RP}^2], and more generally, real projective spaces \mathbb{RP}^n generate in dimensions n \equiv 2,4 \pmod{8} (such as [\mathbb{RP}^4] and [\mathbb{RP}^8] in dimensions 4 and 8). In dimension 4, the two basis elements are [\mathbb{RP}^4] and [\mathbb{CP}^2]; in dimension 5, the generator is [\mathbb{S}^1 \times \mathbb{RP}^4] or the Dold manifold; in dimension 6, basis elements include [\mathbb{RP}^6], [\mathbb{RP}^2 \times \mathbb{RP}^4], and [\mathbb{CP}^3]; in dimension 9, a generator is the 9-dimensional Dold manifold; in dimension 10, representatives include products like \mathbb{RP}^2 \times \mathbb{RP}^8 and others. The full list of groups up to dimension 10 is as follows:| Dimension n | \Omega_n^U |
|---|---|
| 0 | \mathbb{Z}/2 |
| 1 | 0 |
| 2 | \mathbb{Z}/2 |
| 3 | 0 |
| 4 | (\mathbb{Z}/2)^2 |
| 5 | \mathbb{Z}/2 |
| 6 | (\mathbb{Z}/2)^3 |
| 7 | 0 |
| 8 | (\mathbb{Z}/2)^5 |
| 9 | \mathbb{Z}/2 |
| 10 | (\mathbb{Z}/2)^7 |
Oriented Cobordism
Oriented cobordism classifies compact oriented smooth manifolds up to oriented cobordism, where two d-dimensional oriented manifolds M and N are equivalent if there exists a compact oriented (d+1)-dimensional manifold W whose boundary is diffeomorphic to the disjoint union M \sqcup (-N), with -N denoting N with reversed orientation. The set of such equivalence classes forms the abelian group \Omega^{\mathrm{SO}}_d under disjoint union, and the collection \Omega^{\mathrm{SO}}_* = \bigoplus_{d \geq 0} \Omega^{\mathrm{SO}}_d assembles into a graded commutative ring with multiplication induced by the Cartesian product of manifolds.[45][46] In their seminal 1964 work, Conner and Floyd established the structure of the oriented cobordism ring \Omega^{\mathrm{SO}}_*, showing that it is generated as a ring by the classes of even-dimensional complex projective spaces [\mathbb{CP}^{2k}] for k \geq 1. Over the rationals, \Omega^{\mathrm{SO}}_* \otimes \mathbb{Q} is a polynomial algebra \mathbb{Q}[x_4, x_8, x_{12}, \dots ], where each x_{4k} is represented by a rational multiple of [\mathbb{CP}^{2k}]. Integrally, the groups are free abelian in even dimensions, with \Omega^{\mathrm{SO}}_{4k} free abelian of rank p(k), where p(k) is the partition function (number of ways to write k as sum of positive integers disregarding order), generated by products of these projective spaces, while odd-dimensional groups contain 2-torsion arising from the image of the J-homomorphism J: \pi_{n+1}^{\mathrm{St}}(SO) \to \Omega^{\mathrm{SO}}_n.[45][47] The low-dimensional oriented cobordism groups up to dimension 10 are:| Dimension n | \Omega_n^{\mathrm{SO}} |
|---|---|
| 0 | \mathbb{Z} |
| 1 | 0 |
| 2 | 0 |
| 3 | 0 |
| 4 | \mathbb{Z} |
| 5 | \mathbb{Z}/24 |
| 6 | 0 |
| 7 | 0 |
| 8 | \mathbb{Z}^2 \oplus \mathbb{Z}/2 |
| 9 | \mathbb{Z}/2 \oplus \mathbb{Z}/3 |
| 10 | \mathbb{Z}^2 |