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Differential topology

Differential topology is the study of smooth manifolds and the differentiable mappings between them, focusing on topological properties that are invariant under diffeomorphisms. It integrates tools from , such as derivatives and smooth functions, with topological concepts to examine the global and local structures of these spaces in a coordinate-independent manner. A smooth manifold is a that is locally homeomorphic to \mathbb{R}^n and equipped with a smooth atlas, consisting of charts where transition maps are C^\infty diffeomorphisms. These manifolds generalize familiar objects like spheres, tori, and projective spaces, allowing for the definition of smooth maps, tangent vectors, and vector fields in a way that preserves differentiability. Central to the field are notions like immersions (locally injective smooth maps), embeddings (injective immersions that are homeomorphisms onto their image), and diffeomorphisms (bijective smooth maps with smooth inverses), which classify manifolds up to smooth equivalence. Differential topology bridges , which studies continuous deformations, and , which incorporates metric structures like curvature, by emphasizing smooth invariants such as classes and dimensions. Key theorems include , which states that the set of critical values of a map has measure zero, enabling the study of regular values and submanifolds; the transversality theorem, which guarantees that generic maps intersect submanifolds properly; and Whitney's embedding theorem, asserting that any n-dimensional manifold embeds in \mathbb{R}^{2n+1}. These results underpin applications in (e.g., and ), physics (e.g., on curved spacetimes), and even exotic phenomena like sphere eversions and the existence of exotic structures on \mathbb{R}^4. Historically, the field emerged in the early through works by on manifold classification and later advancements by on critical points and vector fields, culminating in foundational theorems by Hassler Whitney and others in the mid-20th century. Today, it remains vital for understanding smooth structures in higher dimensions and intersects with , , and , influencing areas from to .

Introduction

Definition and Scope

Differential topology is the study of smooth manifolds and the properties of these manifolds that remain invariant under , which are bijective smooth maps with smooth inverses. It focuses on classifying manifolds up to diffeomorphism type, meaning two manifolds are considered equivalent if there exists a diffeomorphism between them. This field examines qualitative features that persist through smooth deformations, distinguishing it from more rigid classifications in other areas of . The scope of differential topology encompasses global topological invariants, such as types that capture the essential connectivity of spaces, properties that determine how manifolds can be realized without self-intersections in higher-dimensional spaces, and the structure of groups that describe the symmetries of manifolds. A notable challenge within this scope arises in 4, where exotic structures exist on \mathbb{R}^4; these are manifolds homeomorphic to the standard Euclidean 4-space but not to it, highlighting the subtlety of classification in this . Foundational tools in differential topology include smooth structures on manifolds, which equip topological spaces with compatible smooth atlases, enabling the definition of smooth maps. Key among these are immersions, where the of a map is injective at every point, allowing local embeddings without folds; submersions, where the is surjective, facilitating local projections; and transversality, a condition ensuring that the images of maps intersect cleanly, avoiding tangencies and enabling generic approximations. Differential topology bridges pure and by applying calculus-based techniques, such as and approximations, to investigate qualitative topological properties without imposing a or Riemannian on the manifolds.

Historical Development

The development of differential topology as a distinct field emerged from early 20th-century advances in the of smooth functions on manifolds. In the 1920s, initiated the study of critical points of real-valued functions on manifolds, establishing a variational framework that linked local extrema to global topological features, as detailed in his 1925 paper on relations between critical points. This work, expanded through the and , provided essential precursors for later topological applications, influencing the understanding of manifold without direct concerns. The 1930s marked a pivotal shift with Hassler 's foundational contributions to smooth manifold theory. In his 1936 paper, Whitney proved that any n-dimensional manifold can be embedded as a closed of \mathbb{R}^{2n}, resolving key questions about representing abstract manifolds in familiar Euclidean settings and enabling rigorous study of differentiable structures. This embedding theorem, alongside Whitney's earlier work on triangulations and extensions of differentiable functions, solidified the framework for manifolds during the 1930s and 1940s, bridging and . Post-World War II advancements accelerated the field's growth. René Thom's 1954 introduction of cobordism theory classified oriented smooth manifolds up to bordism using homotopy groups of Thom spaces, providing powerful invariants and sparking applications in . The term "differential topology" began to appear around 1960, coinciding with major breakthroughs: John Milnor's 1956 discovery of exotic smooth structures on the 7-sphere demonstrated that topological spheres could admit multiple non-diffeomorphic smoothings, challenging assumptions about uniqueness in higher dimensions. That same year, Milnor's work highlighted at least seven distinct smooth structures on S^7, later connected to the 28-dimensional group of exotic 7-spheres. Further milestones in the early 1960s underscored the field's maturity. Michel Kervaire's 1960 construction of a 10-dimensional piecewise-linear manifold that admits no smooth structure provided the first example of non-smoothability, revealing discrepancies between smooth, PL, and topological categories. Stephen Smale's 1961 h-cobordism theorem established that simply connected h-cobordisms in dimensions at least 5 are diffeomorphic to products, proving the higher-dimensional and enabling techniques for manifold classification. These results, building on Thom's , positioned differential topology as an independent discipline by the mid-1960s. In the 1980s, Simon Donaldson's application of revolutionized the study of 4-manifolds, using moduli spaces of anti-self-dual connections to derive that distinguished structures and disproved longstanding conjectures, such as the sufficiency of the Rokhlin invariant for 4-manifolds. His 1983 work introduced Donaldson polynomials as , profoundly impacting . By the late 20th and early 21st centuries, efforts persisted on the 4-dimensional , which remains open despite advances in and ; notable progress includes detailed expositions by John Morgan and Gang Tian on related geometric flows, though a full resolution eludes researchers as of 2025.

Foundational Concepts

Smooth Manifolds

A manifold is a endowed with a , defined as an of atlases where the transition maps between charts are infinitely differentiable (C^\infty). A is a Hausdorff, that is locally homeomorphic to \mathbb{R}^n for some fixed dimension n, but the transition maps in its atlas are merely continuous. In contrast, piecewise linear (PL) manifolds require transition maps to be piecewise linear homeomorphisms, providing a structure intermediate between topological and . The allows for the performance of on the manifold, enabling definitions of derivatives, tangent vectors, and differential forms in a coordinate-independent manner. Smooth manifolds are constructed via compatible charts covering the space. For instance, \mathbb{R}^n itself serves as the prototypical smooth n-manifold with the of open sets and identity maps. The n-sphere S^n = \{ x \in \mathbb{R}^{n+1} \mid \|x\| = 1 \} admits a through charts, excluding the and using inverse projections as transition maps, which are rational functions. The n-torus T^n = S^1 \times \cdots \times S^1 (n times) inherits a as a product manifold, with charts derived from angular coordinates on each factor. A maximal atlas is obtained by including all charts compatible with a given atlas, ensuring the structure is uniquely determined up to equivalence. Key properties of smooth manifolds include their dimension n, which is locally constant and globally fixed; , determined by the existence of an atlas where all transition maps have positive determinants; and , inherited from the underlying (e.g., compact manifolds like S^n are bounded and closed in embeddings). The TM is a of rank n over M, with fiber T_pM at each point p \in M consisting of equivalence classes of curves through p under reparametrization, or equivalently, derivations on the space of smooth functions at p. Every smooth manifold admits a , which captures the local linear approximations and is central to defining vector fields and differential operators. Every of 1, 2, or 3 admits a , and this structure is unique up to . However, in higher dimensions, not every topological manifold admits a smooth structure; for example, certain compact 10-dimensional manifolds constructed by Kervaire do not. Moreover, in 4, the topological space \mathbb{R}^4 admits uncountably many distinct structures, known as , which are homeomorphic but not diffeomorphic to the standard smooth \mathbb{R}^4.

Diffeomorphisms and Smooth Maps

In differential topology, a smooth map between smooth manifolds is a function f: M \to N that is C^\infty-differentiable, meaning that in local coordinate charts, the representation of f is infinitely differentiable. More precisely, for charts (U, \Phi) on M and (V, \Psi) on N, the composition \Psi \circ f \circ \Phi^{-1} is a C^\infty map between open subsets of spaces. A is a smooth bijection f: M \to N whose inverse f^{-1}: N \to M is also smooth, serving as an in the of smooth manifolds. Diffeomorphisms preserve the smooth structure, dimension, and local topological properties of manifolds, such as orientability and the topology of tangent spaces. They act as symmetries that classify manifolds up to smooth equivalence, meaning two manifolds are diffeomorphic if there exists such a map between them, which identifies their intrinsic differential topological features. For instance, the surface of a coffee cup and a donut (torus) are diffeomorphic, as both can be smoothly deformed into each other while preserving their one-handled genus. The group \mathrm{Diff}(M) consists of all diffeomorphisms of a manifold M under composition, forming an infinite-dimensional topologized by the . It encodes the symmetries of M and plays a central role in classifying structures, with its B\mathrm{Diff}(M) used to study bundles modeled on M. Special types of include immersions, submersions, and embeddings, which are defined via properties of the df: TM \to TN. An is a where df_p: T_p M \to T_{f(p)} N is injective at every point p \in M, ensuring the map is locally like an inclusion of tangent spaces./10.16/submfd.pdf) A submersion is a where df_p is surjective at every point, implying that level sets are submanifolds of equal to \dim N./10.16/submfd.pdf) An is an that is also a onto its image, providing a proper topological of M into N as a ./10.16/submfd.pdf) For example, the standard of the sphere S^n into \mathbb{R}^{n+1} is an ./10.16/submfd.pdf)

Core Theorems and Tools

is a fundamental result in differential topology that asserts the set of critical values of a smooth map between manifolds has zero in the target manifold. Specifically, let f: M \to N be a smooth map between smooth manifolds of dimensions m and n, respectively. A point x \in M is a critical point if the df_x: T_x M \to T_{f(x)} N has rank less than \min(m, n), and the image of the set of critical points under f is the set of critical values. The theorem states that this set has measure zero in N. More generally, for maps from Euclidean domains, if m > n, the result holds provided the map is of class C^{m-n+1}. The proof proceeds by reducing the manifold case to the setting via charts and embeddings, then using on the source m. For a map f: U \subset \mathbb{R}^m \to \mathbb{R}^n with U open, define the critical set C = \{ x \in U \mid \rank(df_x) < n \}. Partition C into subsets where exactly the first k partial derivatives vanish, for k = 0, \dots, m-1, and handle higher-order critical points separately. For points where the rank deficiency is low, apply the inverse function theorem after a change of coordinates to project the image onto a lower-dimensional slice, using Fubini's theorem to show the image has measure zero by . For high-order critical points (where all derivatives up to order k vanish with k+1 > m/n), expansions around such points bound the volume of the image of small cubes by O(\delta^{m - n(k+1)}), which tends to zero as the cube size \delta \to 0 since the exponent is negative; subdividing domains and summing over subcubes yields measure zero overall. This measure-zero property implies that almost every point in N (in the Lebesgue sense) is a regular value of f, meaning the preimage f^{-1}(y) is either empty or a of M of m - n for y. Consequently, smooth maps can be generically perturbed to achieve desired properties, as the set of "bad" values avoiding transversality has measure zero and can be evaded by small adjustments. The theorem was originally proved by Arthur Sard in 1942 for maps between Euclidean spaces of sufficiently high smoothness. Stronger versions, applicable to maps of finite differentiability class C^k where k depends on the dimension difference (e.g., k \geq m - n + 1), were established by Herbert Federer in his comprehensive treatment of .

Transversality and Morse Theory

In differential topology, transversality is a fundamental concept that ensures intersections between submanifolds occur in a "" manner, avoiding degenerate cases. Specifically, two submanifolds Y and Z of a manifold X are said to intersect transversally at a point p \in Y \cap Z if the spaces satisfy T_p Y + T_p Z = T_p X; this condition implies that the intersection is itself a submanifold of \dim Y + \dim Z - \dim X. For maps f: M \to N between manifolds and a submanifold S \subset N, the map f is transverse to S if for every p \in f^{-1}(S), the image of the df_p(T_p M) + T_{f(p)} S = T_{f(p)} N. This property extends the submanifold definition to mappings and is crucial for studying and embeddings. The transversality theorem asserts that transverse maps are generic in the space of smooth maps, meaning that for fixed manifolds M, N and S \subset N, the set of smooth maps f: M \to N transverse to S is dense and open in the C^\infty-; moreover, by applying to appropriate jet projections, one can show that such maps form a residual set (dense G_\delta) in the space of all smooth maps. This result, originally due to , enables the perturbation of maps to achieve desired intersection properties without altering the homotopy type, facilitating proofs in and theorems. Morse theory provides a powerful framework for analyzing the of manifolds through the critical points of . A f: M \to \mathbb{R} on a compact manifold M is called a Morse if all its critical points are non-degenerate, meaning that the of second derivatives at each critical point p (where df_p = 0) is invertible. The index of a critical point p is defined as the dimension of the negative eigenspace of the at p, which classifies the local behavior: index 0 corresponds to local minima, index \dim M to maxima, and intermediate indices to saddles. By attaching cells to the manifold based on these indices via the gradient flow of f, reconstructs M as a CW-complex whose cells reflect the critical points. A cornerstone of Morse theory is the Morse inequalities, which relate the number of critical points to the topology of M via its Betti numbers b_k = \dim H_k(M; \mathbb{Z}), the ranks of the homology groups. The weak Morse inequalities state that for each k, \sum_{i=0}^k (-1)^i c_i \geq \sum_{i=0}^k (-1)^i b_i, where c_i is the number of critical points of index i. The full alternating sum always satisfies the equality \sum_k (-1)^k c_k = \sum_k (-1)^k b_k = \chi(M), the Euler characteristic of M. \sum_{k=0}^{\dim M} (-1)^k c_k = \sum_{k=0}^{\dim M} (-1)^k b_k These relations quantify how the critical points "generate" the homology of M. Reeb's theorem applies Morse theory to gradient flows, stating that if a Morse function f: M \to \mathbb{R} on a compact manifold M has exactly two critical points (a global minimum and maximum), then M is diffeomorphic to the sphere S^{\dim M}; the gradient flow lines connect the minimum to the maximum without recrossing levels, yielding a cell decomposition with one 0-cell and one \dim M-cell. For applications to homotopy type, consider the height function on the torus T^2 embedded standardly in \mathbb{R}^3: it is Morse with four critical points (minimum of index 0, two saddles of index 1, maximum of index 2), and the sublevel sets deform via gradient flow to reveal the homotopy equivalence to a CW-complex with one 0-cell, two 1-cells, and one 2-cell, matching the Betti numbers b_0 = 1, b_1 = 2, b_2 = 1. This illustrates how Morse functions determine the homotopy type through their critical data.

Advanced Results

h-Cobordism Theorem

The h-cobordism theorem, proved by in 1962, provides a powerful criterion for determining when two simply connected manifolds of dimension at least 5 are . Specifically, it states that if W is a compact between two closed simply connected n-manifolds M_0 and M_1 with n \geq 5, and if the inclusions i_k: M_k \hookrightarrow W for k = 0, 1 are homotopy equivalences (making W an *), then there exists a F: M_0 \times [0,1] \to W such that F|_{M_0 \times \{0\}} = \mathrm{id}_{M_0} and F(M_0 \times \{1\}) = M_1. This implies that M_0 and M_1 are . Smale's proof employs a program based on handlebody decompositions derived from , where the W is expressed as a of handles attached along the bottom M_0. By analyzing the equivalence condition, handles of index 1 and n-1 are shown to cancel pairwise, while higher-index handles (2 through n-2) can be isotoped to lie outside lower-index ones, allowing simplification to a trivial product structure via isotopies in the ambient space. This approach leverages transversality to ensure the necessary embeddings and cancellations, avoiding intersections through the Whitney trick, which is valid in dimensions n \geq 5. Among its consequences, the theorem immediately yields a proof of the higher-dimensional , asserting that every simply connected closed smooth n-manifold homotopy equivalent to the n-sphere is to the standard sphere S^n, for n \geq 5. This was established by Smale in 1961 using the h- theorem to classify such manifolds up to . A variant, the s-cobordism theorem, extends the result to non-simply connected manifolds by incorporating torsion to measure obstructions, as proved by Barden, Mazur, and Stallings in the mid-1960s. The theorem also fails in dimension 4, where Donaldson's gauge-theoretic constructions produce smooth h-cobordisms that are not products, highlighting the exceptional nature of four-dimensional topology. Furthermore, it forms the cornerstone of , enabling the in higher dimensions through controlled handle attachments and isotopies, as developed by Kervaire, Milnor, and .

Exotic Spheres and Manifolds

Exotic spheres are smooth manifolds that are homeomorphic to the standard n-sphere S^n but not diffeomorphic to it, representing distinct smooth structures on the same topological space. These structures highlight the distinction between the topological and smooth categories in differential topology, where homeomorphism does not imply diffeomorphism in dimensions greater than or equal to 7. In 1956, John Milnor constructed the first examples of exotic spheres in dimension 7 using framed cobordisms, showing that there exist smooth 7-manifolds homeomorphic to S^7 but not diffeomorphic to the standard smooth structure. These manifolds arise as total spaces of S^3-bundles over S^4 with non-trivial clutching functions, and Milnor demonstrated that at least one such bundle yields an exotic structure by computing its Pontryagin numbers, which differ from those of the standard sphere. Further analysis revealed a total of 28 distinct exotic 7-spheres up to diffeomorphism, forming the group \Theta_7 \cong \mathbb{Z}/28\mathbb{Z}. The Kervaire-Milnor classification provides a complete description of exotic spheres in dimensions n = 4k-1 by identifying the group of homotopy n-spheres \Theta_n with a quotient of the homotopy groups of spheres, specifically \Theta_n \cong \pi_{n}^S / bP_{n}, where bP_n is the image of the Bernoulli numbers under the J-homomorphism. This classification shows that exotic spheres exist precisely when this group is non-trivial, which occurs in most high dimensions but not in all low ones. No exotic spheres exist in dimensions 1 through 3, 5, or 6, as \Theta_n = 0 in these cases, while dimension 4 remains open for the smooth Poincaré conjecture. Exotic smooth structures also appear on \mathbb{R}^4, with Michael Freedman establishing the existence of topological manifolds homeomorphic to \mathbb{R}^4 but incompatible with the standard smooth structure in the 1980s; subsequent progress, including constructions via gauge theory failures in dimension 4, confirmed infinitely many smooth exotic \mathbb{R}^4s in the mid-1980s. Exotic \mathbb{R}^4s are often classified as "small" (diffeomorphic to the standard \mathbb{R}^4 outside a compact set) or "large" (not embeddable in the standard \mathbb{R}^4), with ongoing refinements as of 2025 exploring their embeddability, moduli, and diffeomorphisms. These examples underscore the failure of smooth uniqueness in high dimensions, contrasting with the h-cobordism theorem's guarantees for simply connected manifolds of dimension at least 5, and emphasize how differential topology reveals rigidities and pathologies absent in the topological category.

Comparisons and Relations

Versus Differential Geometry

Differential topology and differential geometry both study smooth manifolds, but they diverge in their primary focus and tools. Differential topology examines the global properties of smooth manifolds up to , emphasizing qualitative aspects such as the existence of smooth maps and embeddings without reference to or distances. In contrast, differential geometry equips manifolds with a Riemannian to investigate quantitative features like , geodesics, and local invariants that measure and size. This distinction means differential topology classifies manifolds based on their diffeomorphism type, often ignoring metric structures, while differential geometry analyzes how induce geometric invariants that can distinguish non-diffeomorphic spaces. A key example illustrates this contrast: in differential topology, all smooth manifolds of the same dimension are locally diffeomorphic to , treating them as qualitatively similar in their infinitesimal structure without regard to embedded . However, differential geometry differentiates them through invariants like ; for instance, the 2-sphere with its standard has positive , while the 2-torus with a flat has zero, reflecting distinct geometric properties despite potential topological similarities. Another illustrative case is the circle: all smooth embeddings of the 1-sphere are diffeomorphic, disregarding radius or in , whereas quantifies these via the , such as or distances. represents an overlap, where manifolds carry a compatible form—a closed, non-degenerate 2-form—blending topological classification with geometric constraints on flows. Historically, originated in the with foundational work by Gauss on surface and Riemann's generalization to abstract manifolds via . evolved from this classical framework in the post-1950s era, shifting emphasis to global smooth equivalence classes through breakthroughs like Smale's h-cobordism theorem and Milnor's discovery of exotic spheres, prioritizing over metric analysis. This transition marked a move from local geometric measurements to broader topological invariants, influencing modern manifold classification.

Versus Algebraic Topology

Differential topology and algebraic topology both study topological invariants of spaces, but they differ fundamentally in their approaches and tools. Algebraic topology primarily employs combinatorial methods, such as CW-complexes and simplicial complexes, to compute groups and groups, focusing on discrete approximations of spaces that capture their global structure without reference to . In contrast, differential topology leverages structures on manifolds, using differential forms and fields to refine these invariants, emphasizing local differentiability and analytic properties that arise from the smooth category. Significant overlaps exist, particularly in how both fields compute key invariants like , yet distinctions emerge in their proofs and interpretations. For instance, in is established through cellular chains on triangulated manifolds, pairing classes via intersection numbers in the chain complex. In differential topology, this duality is realized via with smooth differential forms, where the on corresponds to wedge products of forms, providing a more geometric perspective on manifold orientations and dimensions. A notable example is the , which asserts that no continuous nowhere-vanishing exists on the even-dimensional ; its differential topology proof links the index of vector fields to the , offering an analytic demonstration of a purely topological obstruction absent in algebraic treatments. The de Rham theorem provides a profound bridge between the two fields, establishing an isomorphism between the of a smooth manifold—computed using closed differential forms modulo exact ones—and the from , ensuring that smooth invariants align with combinatorial ones under suitable conditions. In modern developments, differential topology extends beyond classical algebraic invariants through , which constructs new polynomial invariants for four-manifolds via moduli spaces of anti-self-dual connections on principal bundles; these Donaldson invariants detect smooth structures and types that algebraic topology alone cannot distinguish, revolutionizing .

Applications

In Physics

Differential topology provides foundational tools for modeling in , where the universe is described as a smooth manifold equipped with a that encodes gravitational effects. Seminal applications leverage theory to analyze the global structure of spacetimes, particularly obstructions to extending metrics or s across boundaries. For instance, spin-Lorentz cobordisms ensure compatibility between spacetime regions with non-empty boundaries, revealing that the type of the does not necessarily determine the causal structure in compact spacetimes. also appear in formulations of , facilitating topology changes induced by pointlike sources, which contribute to transition amplitudes between distinct spacetime configurations. Exotic smooth structures on spacetime manifolds, such as those on \mathbb{R}^4, introduce alternative differentiable atlases but are often physically irrelevant, as observable physics relies on the compatible with the physical in . In gauge theories, differential topology intersects with Yang-Mills theory to probe the of four-manifolds, yielding invariants that distinguish topological properties. During the 1980s, developed polynomial invariants from the moduli spaces of anti-self-dual Yang-Mills connections on vector bundles over oriented four-manifolds, providing constraints on intersection forms and revealing non-standard . These invariants, derived from the orientation of Yang-Mills moduli spaces, apply to manifolds with arbitrary fundamental groups and negative definite intersection forms, equating them to standard diagonal forms over the integers. Building on this, Seiberg-Witten invariants, introduced in the , simplify computations in supersymmetric theories on four-manifolds, offering stronger constraints on by counting solutions to perturbed equations. These invariants detect differences in versus topological four-manifolds, with applications to Donaldson theory via structures. String theory employs differential topology through Calabi-Yau manifolds, which compactify extra dimensions while preserving supersymmetry, and symplectic topology underpins mirror symmetry relating pairs of such manifolds. Mirror symmetry equates the complex geometry of one Calabi-Yau manifold to the symplectic geometry of its mirror, predicting identical physical spectra and enabling computations of Yukawa couplings via Picard-Fuchs equations on hypersurfaces in weighted projective spaces. Cobordism theory ensures anomaly cancellation in string compactifications, where triviality of cobordism classes in supergravity dimensions greater than six imposes constraints on gauge groups and requires exotic defects like I-folds to realize anomaly-free theories. Up to 2025, topological invariants from differential topology classify microstates in , constructing horizonless geometries that match entropies through duality-invariant charge combinations. Analogs of the theorem emerge in via the swampland conjecture, positing trivial cobordism groups to exclude non-physical global symmetries and enforce stringy defects in effective theories.

In Pure Mathematics

In , differential topology provides essential tools for classifying surfaces through , which relates the critical points of smooth functions on a manifold to its topological structure. For compact orientable surfaces, Morse functions allow the decomposition into handles, enabling a complete classification up to based on and , as detailed in the handlebody decomposition. This approach extends to non-orientable surfaces by analogous handle attachments, confirming the classical classification theorem that distinguishes them by and crosscap number. Surgery techniques from differential topology further contribute to low-dimensional topology by constructing knot invariants, particularly through Dehn surgery on knots in three-manifolds, which modifies the manifold to produce new invariants like the or linking numbers. These operations preserve essential topological features while allowing computation of concordance invariants, as surveyed in applications linking surgery to knot complements and their fundamental groups. For instance, surgery along a yields manifolds whose detects non-trivial knotting, providing obstructions to unknotting. In , transversality theorems from differential topology facilitate the by ensuring intersections avoid singular loci during blow-up processes or alterations. Hironaka's theorem on resolution in characteristic zero relies on such perturbations to iteratively resolve singularities via sequences of blow-ups, transforming singular varieties into models while preserving birational . reductions, blending differential topology with algebraic structures, reduce quotient spaces under group actions to symplectic quotients, as in where moment map levels yield stable moduli spaces of varieties. Arithmetic topology draws analogies between differential topology and , modeling prime ideals in rings of integers as s in three-manifolds, with the complement corresponding to the étale of the spec of the . Closed orientable three-manifolds analogize number fields, and invariants like the mirror L-functions of primes, facilitating translations of conjectures such as the Hasse norm principle to manifold properties. This framework, initiated in the , has led to insights into prime distribution via concordance. A landmark application is Perelman's 2003 proof of the using with , a differential topology technique that evolves metrics on three-manifolds to reveal their topological type; by analyzing singularities and performing controlled surgeries, he showed that simply connected closed three-manifolds are homeomorphic to the three-sphere, blending dynamics with topological invariants. This proof, expanded in subsequent verifications, also advanced the . Progress in four-manifold up to 2025 includes topological classifications via Freedman's work on simply connected cases and recent classifications for manifolds with three-manifold fundamental groups, though smooth structures remain elusive beyond Donaldson's invariants. Embeddings in leverage differential topology to study structures, where Legendrian knots and their embeddings into three-manifolds determine overtwisted versus tight structures via h-principle results ensuring generic transversality. These embeddings classify structures on three-manifolds, with symplectic fillings providing obstructions through Weinstein handles. In higher dimensions, symplectic embeddings of manifolds preserve the induced form, aiding the study of exotic structures.