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In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components.

Examples

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The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. The real projective space RPn is a closed n-dimensional manifold. The complex projective space CPn is a closed 2n-dimensional manifold.[1] A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary.

Properties

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Every closed manifold is a Euclidean neighborhood retract and thus has finitely generated homology groups.[2]

If is a closed connected n-manifold, the n-th homology group is or 0 depending on whether is orientable or not.[3] Moreover, the torsion subgroup of the (n-1)-th homology group is 0 or depending on whether is orientable or not. This follows from an application of the universal coefficient theorem.[4]

Let be a commutative ring. For -orientable with fundamental class , the map defined by is an isomorphism for all k. This is the Poincaré duality.[5] In particular, every closed manifold is -orientable. So there is always an isomorphism .

Open manifolds

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For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.

Abuse of language

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Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical conditions), thus by this definition a manifold does not include its boundary when it is embedded in a larger space. However, this definition doesn’t cover some basic objects such as a closed disk, so authors sometimes define a manifold with boundary and abusively say manifold without reference to the boundary. But normally, a compact manifold (compact with respect to its underlying topology) can synonymously be used for closed manifold if the usual definition for manifold is used.

The notion of a closed manifold is unrelated to that of a closed set. A line is a closed subset of the plane, and it is a manifold, but not a closed manifold.

Use in physics

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The notion of a "closed universe" can refer to the universe being a closed manifold but more likely refers to the universe being a manifold of constant positive Ricci curvature.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

Closed manifold

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In , a closed manifold is a manifold without boundary, defined as a Hausdorff, second-countable that is locally homeomorphic to Rn\mathbb{R}^n for some fixed dimension n>0n > 0. This structure ensures the space is "finite" in a global sense due to compactness, while locally resembling flat space without any edges or boundaries. Closed manifolds are central to algebraic topology and differential geometry, providing models for spaces with well-defined global invariants like the Euler characteristic χ(M)\chi(M), which for an orientable closed nn-manifold equals the alternating sum of Betti numbers from its homology groups. They support additional structures, such as smooth atlases for differentiable manifolds or Riemannian metrics for geometric studies, and satisfy Poincaré duality: for an orientable closed nn-manifold MM, the cohomology group Hk(M;R)H^k(M; \mathbb{R}) is isomorphic to the homology group Hn−k(M;R)H_{n-k}(M; \mathbb{R}). Orientability, detected by whether the top homology Hn(M;Z)≅ZH_n(M; \mathbb{Z}) \cong \mathbb{Z} or 00, distinguishes manifolds like the sphere from the real projective plane. Classic examples include the nn-sphere SnS^n, with homology concentrated in degrees 0 and nn (H0(Sn;Z)≅ZH_0(S^n; \mathbb{Z}) \cong \mathbb{Z}, Hn(Sn;Z)≅ZH_n(S^n; \mathbb{Z}) \cong \mathbb{Z}, and zero elsewhere); the nn- Tn=(S1)nT^n = (S^1)^n, featuring free abelian homology Hk(Tn;Z)≅Z(nk)H_k(T^n; \mathbb{Z}) \cong \mathbb{Z}^{\binom{n}{k}}; and the real RPn\mathbb{RP}^n, which is non-orientable for even nn with twisted homology (e.g., H1(RPn;Z)≅Z/2ZH_1(\mathbb{RP}^n; \mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z} for n≥2n \geq 2). In dimension 2, closed manifolds are classified up to : orientable ones as connected sums of tori (genus g≥0g \geq 0), with χ=2−2g\chi = 2 - 2g, and non-orientable ones as connected sums involving projective planes. Higher-dimensional cases, such as simply connected closed 3-manifolds, were resolved by the (proved by Perelman in 2003), asserting they are homeomorphic to S3S^3. These objects underpin theorems like the Brouwer fixed-point theorem and van Kampen's theorem, influencing fields from to physics via models like configuration spaces.

Definition and Fundamentals

Formal Definition

A closed manifold is an nn-dimensional MnM^n that is compact and has empty boundary ∂M=∅\partial M = \emptyset. A is a Hausdorff, second-countable that is locally Euclidean: for each point p∈Mp \in M, there exists an open neighborhood U⊂MU \subset M containing pp and a ϕ:U→Rn\phi: U \to \mathbb{R}^n to an open subset of Rn\mathbb{R}^n, where nn is the dimension of MM. The space is second-countable to ensure it has a countable basis for its topology, which implies paracompactness and allows for the existence of partitions of unity in more advanced settings. Compactness means that every open cover of MM admits a finite subcover. The absence of boundary requires that every maps to an open subset of Rn\mathbb{R}^n, with no points whose neighborhoods are homeomorphic only to the half-space Hn={(x1,…,xn)∈Rn∣xn≥0}\mathbb{H}^n = \{ (x_1, \dots, x_n) \in \mathbb{R}^n \mid x_n \geq 0 \}. Closed manifolds exist in dimensions n≥0n \geq 0; for n=0n=0, they are finite discrete sets of points, as R0\mathbb{R}^0 is a single point and restricts the to finitely many isolated points.

Distinction from General Manifolds

In general, a manifold is a that is locally homeomorphic to Rn\mathbb{R}^n, typically assumed to be Hausdorff and second countable to ensure desirable properties like paracompactness. However, manifolds can exhibit a wide range of global behaviors: they may be non-compact, such as Rn\mathbb{R}^n itself, which extends infinitely without bound; they may possess a boundary, as in the case of the closed unit disk DnD^n, where points on the boundary have neighborhoods homeomorphic to half-spaces Hn\mathbb{H}^n; or, in less standard constructions, they may fail to be Hausdorff, though such examples are rare in mainstream topology. A closed manifold, by contrast, satisfies all the local Euclidean conditions while imposing strict global constraints: it must be compact and without boundary, ensuring ∂M=∅\partial M = \emptyset. The key distinctions arise from these global prerequisites. Non-compact manifolds like Rn\mathbb{R}^n fail to be closed because they lack ; for instance, the open cover by balls of radius kk for k∈Nk \in \mathbb{N} has no finite subcover, reflecting their unbounded nature. Manifolds with boundary, such as the disk DnD^n, are compact but not closed since their boundary is nonempty, introducing points where local structure deviates from full . Moreover, while closed manifolds are inherently topological (requiring only homeomorphisms), more structured variants like piecewise linear (PL) or differentiable manifolds are not necessary for the basic notion of closedness, though they often appear in applications. These differences highlight that closedness is not merely a topological but a defining structural feature. The implications of these prerequisites are profound for the and of closed manifolds. guarantees a form of finite "size," meaning the manifold can be covered by finitely many charts, which facilitates techniques like exhaustion by compact subsets and ensures integrals or sums over the space converge without additional assumptions. The absence of a boundary implies a fully "internal" , where every point admits a neighborhood diffeomorphic (or homeomorphic, in the topological case) to an open ball in Rn\mathbb{R}^n, avoiding edge effects that complicate computations in manifolds with boundary. Together, these ensure closed manifolds behave as self-contained objects ideal for studying global invariants. Historically, the term "closed manifold" (from the French "variétés fermées") originates in , introduced by in his foundational 1895 paper "Analysis Situs," where it distinguished compact manifolds without boundary from open subsets or incomplete surfaces in the study of homology and connectivity. This usage contrasted with "open" manifolds, emphasizing the topological closure properties essential for cycle decompositions and duality theorems.

Examples and Classifications

Classic Examples

In low dimensions, the standard examples of closed manifolds illustrate the core features of and absence of boundary. The 0-dimensional S0S^0, consisting of the two points {−1,1}⊂R\{ -1, 1 \} \subset \mathbb{R}, serves as the simplest closed 0-manifold; it is compact as a and has no boundary since points in 0 dimensions inherently lack edges. In one dimension, the circle S1={(x,y)∈R2∣x2+y2=1}S^1 = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1 \} is the unique connected closed 1-manifold up to , compact as a bounded closed subset of R2\mathbb{R}^2 and boundary-free by its smooth embedding without endpoints. Two-dimensional closed manifolds, or closed surfaces, provide a rich set of classic examples. The 2-sphere S2={(x,y,z)∈R3∣x2+y2+z2=1}S^2 = \{ (x,y,z) \in \mathbb{R}^3 \mid x^2 + y^2 + z^2 = 1 \} is the prototypical closed orientable surface of 0, compact via its construction and without boundary as a smooth hypersurface. The T2=S1×S1T^2 = S^1 \times S^1 represents the closed orientable surface of 1, obtained as a product of compact spaces, hence compact, with no boundary due to the boundary-free nature of each factor. The real RP2\mathbb{RP}^2, formed as the S2/∼S^2 / \sim where antipodal points are identified, is a classic closed non-orientable surface; its compactness follows from the finite identification on the compact S2S^2, and it admits no boundary as a smooth manifold. Closed surfaces admit a complete classification up to . All closed orientable surfaces are diffeomorphic to with gg handles attached, for g=0,1,2,…g = 0, 1, 2, \dots, where gg is the ; for instance, g=0g=0 yields S2S^2, g=1g=1 the , and higher gg produce increasingly complex surfaces like the double torus, all compact and boundary-free by construction from polygonal identifications or handle attachments. Non-orientable closed surfaces include the (RP2\mathbb{RP}^2, one cross-cap) and the , obtained via specific identifications on a square (e.g., opposite sides glued with twists); the is compact as a of the compact square and lacks a boundary in its manifold . In higher dimensions, the n-spheres Sn={x∈Rn+1∣∥x∥=1}S^n = \{ x \in \mathbb{R}^{n+1} \mid \|x\| = 1 \} for n≥3n \geq 3 generalize the lower-dimensional cases, remaining closed n-manifolds that are compact subsets of Euclidean space and smooth without boundary. Real projective spaces RPn=Sn/{±1}\mathbb{RP}^n = S^n / \{\pm 1\}, quotients of SnS^n by the antipodal map, form closed n-manifolds; compactness arises from the continuous action of a compact group on a compact space, with no boundary inherited from SnS^n. Complex projective spaces CPn\mathbb{CP}^n, constructed as quotients S2n+1/S1S^{2n+1} / S^1 where S1S^1 acts by complex multiplication, yield closed 2n-manifolds; they are compact due to the quotient of the compact odd-dimensional sphere by a compact Lie group and without boundary, inheriting the closed structure from the quotient construction.

Counterexamples and Edge Cases

While Euclidean space Rn\mathbb{R}^n for n≥1n \geq 1 satisfies the local Euclidean and Hausdorff conditions of a topological manifold, it fails to be closed because it is not compact; sequences can escape to infinity without converging. Similarly, the punctured torus, obtained by removing a single point from the standard closed 2-torus T2T^2, is an open manifold that is non-compact and thus not closed, as it inherits the local Euclidean structure but loses global compactness. Manifolds with nonempty boundary provide another key counterexample, as closed manifolds require the boundary to be empty. The closed nn-dimensional disk Dn={x∈Rn:∥x∥≤1}D^n = \{ x \in \mathbb{R}^n : \|x\| \leq 1 \} is compact and locally Euclidean (with interior points modeled on open balls and boundary points on half-spaces), but its boundary ∂Dn=Sn−1\partial D^n = S^{n-1} is nonempty, disqualifying it as closed. The , a non-orientable surface formed by identifying the ends of a rectangular strip with a twist, is likewise a 2-manifold with boundary (a single circle) and is compact, yet the presence of this boundary prevents it from being closed. In one dimension, the closed interval [0,1][0,1] serves as a basic example: it is compact with boundary points at 0 and 1, each requiring half-open interval charts, but the nonempty boundary ∂[0,1]={0,1}\partial [0,1] = \{0,1\} excludes it from the closed category. Edge cases further illustrate failures in the standard axioms for manifolds, which closed manifolds must satisfy. An uncountable , such as the set of real numbers with the , is locally Euclidean (each point is an open 0-ball) but not second-countable, as it requires uncountably many singleton open sets for a basis, violating the paracompactness and metrizability implicit in manifold theory. Non-Hausdorff examples, like the line with two origins—constructed as the quotient of two copies of R\mathbb{R} glued along R∖{0}\mathbb{R} \setminus \{0\}—are locally Euclidean but fail the , as the two origin points cannot be separated by disjoint open neighborhoods; such spaces are sometimes called "exotic manifolds" but do not qualify as standard closed manifolds due to this topological defect.

Key Properties

Topological Characteristics

Closed manifolds, being compact without boundary, exhibit several fundamental topological properties arising from their . In particular, any continuous of a closed n-manifold MM into RN\mathbb{R}^N yields a closed and bounded , as ensures the is compact and hence satisfies the Heine-Borel property. Moreover, implies that MM is equivalent to a finite CW-complex, so its groups Hk(M;Z)H_k(M; \mathbb{Z}) are finitely generated abelian groups, with finite ranks known as the Betti numbers bk(M)=\rankHk(M;Q)b_k(M) = \rank H_k(M; \mathbb{Q}). This finiteness extends to other global invariants, such as the , defined by the alternating sum χ(M)=∑k=0n(−1)kbk(M),\chi(M) = \sum_{k=0}^n (-1)^k b_k(M), which is thus a well-defined finite for any closed n-manifold MM. Orientability is another key topological feature, though not all closed manifolds possess it; for example, the real RP2\mathbb{RP}^2 provides a compact non- 2-manifold without boundary. For the specific case of closed surfaces (2-dimensional manifolds), these are fully classified up to by two invariants: their and gg, where gg is a non-negative representing the maximal number of disjoint simple closed curves that can be removed without disconnecting the surface. closed surfaces of gg have χ=2−2g\chi = 2 - 2g, while non- ones have χ=2−g\chi = 2 - g. For closed orientable n-manifolds, establishes a profound in the , stating that the groups satisfy an Hk(M;R)≅Hn−k(M;R)H^k(M; \mathbb{R}) \cong H_{n-k}(M; \mathbb{R}) for real coefficients, with analogous statements holding over other fields or for integer coefficients up to torsion. This duality implies bk(M)=bn−k(M)b_k(M) = b_{n-k}(M), leading to χ(M)=0\chi(M) = 0 for odd-dimensional cases, as the alternating sum pairs terms that cancel. In even dimensions congruent to 2 modulo 4 (i.e., n=4m+2n = 4m + 2), the is even, a consequence of the pairing structure in the intersection form on middle-dimensional . further ensures that groups πk(M)\pi_k(M) are finitely generated for k≤nk \leq n, though they may be infinite in general; for instance, the n-sphere [Sn](/page/N−sphere)[S^n](/page/N-sphere), a prototypical closed manifold, has finite groups except in degree n. A significant embedding property is that every closed topological n-manifold admits a topological as a closed subset into some RN\mathbb{R}^N for sufficiently large NN, with N=2n+1N = 2n + 1 often sufficient in high dimensions. This result follows from general theorems for embedding compact manifolds into .

Analytic and Geometric Features

Closed manifolds, when endowed with a , exhibit rich analytic and geometric properties that stem from their . In dimensions 1, 2, and 3, every topological closed manifold admits a unique smooth structure up to , enabling the study of differentiable functions and tensors. In higher dimensions, while not all topological closed manifolds admit smooth structures—counterexamples exist starting from dimension 4, such as certain 4-manifolds resolved by Freedman's work—the h-cobordism theorem, proved by Smale for dimensions greater than or equal to 5, plays a crucial role in understanding the equivalence of smooth structures on simply connected manifolds through arguments. This theorem implies that h-cobordant smooth manifolds are diffeomorphic, facilitating the classification of smooth structures on closed manifolds in many cases. Every smooth closed manifold admits a , which can be constructed using a subordinate to a finite atlas of coordinate charts, where local Euclidean metrics are extended smoothly. On such a (M,g)(M, g), ensures that the metric is complete, and by the Hopf-Rinow , the manifold is geodesically connected: any two points can be joined by a minimizing segment. Moreover, the Lyusternik–Fet guarantees the of at least one closed on any smooth closed of dimension at least 2, reflecting the periodic nature of geodesics due to . These geodesics are critical points of the energy functional on the loop space and play a key role in variational . The finite volume of a closed , a direct consequence of , implies that geometric invariants like total are well-defined and finite. For a closed orientable surface MM equipped with a Riemannian metric gg, the relates the total to the : ∫MKg dAg=2πχ(M),\int_M K_g \, dA_g = 2\pi \chi(M), where KgK_g is the and dAgdA_g the area element; this holds for any metric on the surface and underscores the topological invariance of the . In higher dimensions, for a closed Riemannian nn-manifold (M,g)(M, g), the total is finite: ∫MScalg dvolg<∞,\int_M \mathrm{Scal}_g \, d\mathrm{vol}_g < \infty, since the scalar curvature Scalg\mathrm{Scal}_g is continuous on the compact set MM, and the volume vol(M)\mathrm{vol}(M) is finite. These finite integrals provide bounds on geometric quantities and are essential in theorems like Myers' theorem on diameter and injectivity radius. Additionally, the fundamental group π1(M)\pi_1(M) of a closed manifold MM is finitely presented, as MM has the homotopy type of a finite CW-complex. This finiteness ensures that π1(M)\pi_1(M) admits finite-sheeted covering spaces corresponding to finite-index subgroups, which inherit many geometric properties from MM, such as the existence of Riemannian metrics with controlled curvature. Such covers are instrumental in studying spectral geometry and rigidity theorems on closed manifolds.

Comparative Concepts

Relation to Open Manifolds

Open manifolds are manifolds without boundary that are non-compact, meaning they are not contained within any compact subset of themselves. Examples include Rn\mathbb{R}^n and punctured spaces such as Rn∖{0}\mathbb{R}^n \setminus \{0\}. In contrast to closed manifolds, which are compact and thus have finite diameter and can be covered by finitely many coordinate charts, open manifolds extend infinitely and may require infinitely many charts for a complete covering. A key structural difference arises in their geometric and topological : open manifolds often possess infinite under metrics, such as the Euclidean metric on Rn\mathbb{R}^n, whereas closed manifolds always have finite . Topologically, the homology groups of closed manifolds are finitely generated due to , leading to finite-dimensional vector spaces over fields like Q\mathbb{Q}; open manifolds, however, can have non-finitely generated homology, as seen in certain infinite-genus surfaces. Closed manifolds can be viewed as quotients of open ones by actions that are cocompact, effectively "completing" the space to a finite extent. Open manifolds frequently arise as "ends" of closed ones by removing points or compact subsets; for instance, removing a single point from the nn- SnS^n yields Rn\mathbb{R}^n, which is diffeomorphic to the open manifold obtained via . In cases where an open manifold has a single end and is simply connected at infinity—meaning the fundamental group of the complements of compact subsets stabilizes to the in the —one-point compactification can yield a closed manifold, as in the case of Rn\mathbb{R}^n compactifying to SnS^n. In , the hyperbolic plane H2H^2 serves as a prototypical open manifold of constant negative . Its quotients by torsion-free discrete subgroups of isometries (Fuchsian groups) can yield closed hyperbolic surfaces when the action is cocompact, resulting in finite-area compact manifolds without cusps.

Manifolds with Boundary

A manifold with boundary is defined as a second-countable Hausdorff where every point has a neighborhood homeomorphic to an open subset of the closed half-space Hn={(x1,…,xn)∈Rn∣xn≥0}\mathbb{H}^n = \{(x_1, \dots, x_n) \in \mathbb{R}^n \mid x_n \geq 0\}, equipped with a via a maximal atlas of charts whose transition maps are smooth and compatible with the half-space model. The boundary ∂M\partial M consists of those points whose chart images lie on ∂Hn=Rn−1×{0}\partial \mathbb{H}^n = \mathbb{R}^{n-1} \times \{0\}, and ∂M\partial M itself forms a smooth (n−1)(n-1)-manifold without boundary. In contrast to closed manifolds, which are compact and lack any boundary subset, manifolds with boundary possess a nonempty ∂M\partial M that separates interior points from the exterior. For instance, the closed nn-ball BnB^n is a compact manifold with boundary ∂Bn=Sn−1\partial B^n = S^{n-1}, whereas its interior int⁡(Bn)\operatorname{int}(B^n) is an open manifold homeomorphic to Rn\mathbb{R}^n, highlighting how the boundary imparts compactness to the full space while the interior remains noncompact and boundaryless. Manifolds with boundary admit a collar neighborhood theorem, which guarantees the existence of an open collar C⊂MC \subset M containing ∂M\partial M and diffeomorphic to [0,1)×∂M[0,1) \times \partial M, with {0}×∂M\{0\} \times \partial M identified with ∂M\partial M; closed manifolds, being boundaryless, lack this structure. A key construction relating the two is the double of a compact manifold MM with boundary: form D(M)D(M) by taking two disjoint copies of MM and gluing them along ∂M\partial M via the identity map, yielding a closed (compact boundaryless) nn-manifold. The absence of boundary in closed manifolds has profound implications, particularly in index theory; the Atiyah-Singer index theorem provides a topological formula for the index of elliptic operators on closed manifolds, whereas on manifolds with boundary, the Atiyah-Patodi-Singer variant incorporates boundary corrections via the eta invariant to account for the boundary's spectral contributions.

Terminology and Historical Notes

Common Misuses

One common misuse involves referring to compact manifolds with boundary as "closed," overlooking the precise requirement that closed manifolds must lack a boundary. For instance, the closed unit disk in R2\mathbb{R}^2 is compact but possesses a boundary (the unit ), rendering it a manifold with boundary rather than a closed manifold. This error occasionally extends to projective spaces, where the disk model of the real might be mistakenly labeled closed despite its boundary in that representation, though the actual projective plane is a closed manifold without boundary. Another frequent confusion arises between "closed" and "complete" in the context of Riemannian manifolds, where closed manifolds (being compact without boundary) are automatically complete with respect to the induced metric by the Hopf-Rinow theorem, but completeness does not imply closedness. Rn\mathbb{R}^n, for example, is complete as a Riemannian manifold yet non-compact and thus not closed. This distinction is critical, as non-compact complete manifolds like exhibit unbounded geodesics without qualifying as closed. In , the term "closed" is often misapplied to subsets rather than entire manifolds, such as labeling closed submanifolds (compact embedded submanifolds whose image is a closed subset of the ambient space) as full closed manifolds. A line embedded in R2\mathbb{R}^2 serves as a closed submanifold but fails to be a closed manifold due to its non-compactness and lack of boundary enforcement in the global sense. This abuse blurs the boundary-free compactness inherent to closed manifolds.

Development in Mathematics

The study of closed manifolds originated in the late 19th century with Henri Poincaré's groundbreaking 1895 paper "Analysis Situs," which introduced the abstract concept of manifolds as spaces locally resembling and developed early tools from to analyze their global properties. This work established fundamental invariants, such as homology groups, that distinguish closed manifolds topologically. By 1900, the topological classification of closed surfaces was complete, revealing that orientable ones are determined solely by their , while non-orientable examples like the provide additional cases. Key milestones in the early included Luitzen Brouwer's 1911 , which demonstrated that any continuous map from a closed to itself has a fixed point, offering insights into the behavior of compact spaces related to closed manifolds despite the boundary involvement. In the 1960s, Stephen Smale's h- theorem revolutionized the understanding of smooth structures on closed manifolds of dimension at least 5, showing that homotopy equivalent simply connected manifolds are diffeomorphic under certain conditions, thereby resolving the in high dimensions. A major advance in the classification of closed 3-manifolds came through in the 1980s, which posits that every closed can be decomposed into pieces each admitting one of eight geometric structures, implying a complete . This was spectacularly proved by in the early 2000s using techniques, resolving the longstanding that every simply connected closed is homeomorphic to the S3S^3. Entering the 21st century, research on closed 4-manifolds highlighted the richness of smooth structures, with Michael Freedman's 1982 classification theorem establishing that simply connected closed topological 4-manifolds are determined by their intersection forms. Complementing this, Simon Donaldson's 1983 application of introduced invariants that detect exotic smooth structures, proving the existence of homeomorphic but not diffeomorphic closed 4-manifolds, thus underscoring the exotic nature of dimension 4.

Applications and Implications

Role in Physics

In , spacetime is modeled as a four-dimensional Lorentzian manifold, and certain cosmological models describe with compact closed spatial sections without boundary, ensuring finite spatial volume and no edges. Such configurations align with the no-boundary proposal, where the wave function of the is computed via path integrals over compact geometries lacking an initial boundary. Closed manifolds play a key role in the of in , where the six additional spatial dimensions are compactified on Ricci-flat Kähler manifolds known as Calabi-Yau spaces to preserve and yield effective four-dimensional physics. These compact, closed manifolds ensure the extra dimensions curl up without boundaries, allowing the theory to match observed while hiding the higher-dimensional structure. In , path integrals are often formulated over closed manifolds to define partition functions or amplitudes, particularly in Euclidean signatures where aids convergence and avoids boundary terms. On closed even-dimensional manifolds, certain anomalies, such as purely gravitational ones in four dimensions, vanish due to topological constraints and index theorems, ensuring consistency of the quantum theory without uncanceled contributions. Representative examples include the toroidal universe model, where spatial slices form a three-torus T3T^3, a compact closed manifold compatible with flat or curved metrics in and potentially explaining cosmic microwave background patterns. As of 2025, CMB data analyses suggest possible mild positive spatial curvature consistent with closed topologies, though flat models remain preferred. While black hole event horizons act as internal boundaries separating causal regions, the full in specific closed universe models like the no-boundary proposal is compact without external boundary.

Broader Mathematical Uses

Closed manifolds play a central role in dynamical systems, particularly as compact phase spaces that ensure the existence of invariant measures and support hyperbolic behaviors. Anosov flows, which are a class of uniformly hyperbolic dynamical systems, are well-defined on closed manifolds, where the compactness guarantees the transversality of stable and unstable foliations. For instance, on closed 3-manifolds, such flows can be constructed by gluing filtrating neighborhoods of hyperbolic fixed points, yielding transitive or nontransitive examples with controlled periodic orbits. In ergodic theory, the compactness of closed manifolds facilitates the study of invariant measures and Lyapunov exponents for diffeomorphisms, enabling the application of Pesin theory to non-uniformly hyperbolic systems. This framework, developed for smooth actions on compact manifolds, provides tools to analyze mixing properties and entropy without requiring uniformity in expansion rates. In , closed manifolds admit rich structures that encode global topological invariants through their rings. The ring H∗(M;F2)H^*(M; \mathbb{F}_2) of a closed MM, for example, is characterized by specific relations, such as the structure where higher-degree elements vanish appropriately, reflecting the inherent to orientable closed manifolds. This ring structure aids in distinguishing manifolds up to in low dimensions. Similarly, provides classifications of vector bundles over closed manifolds, with algebraic K-groups of the offering obstructions to stable diffeomorphisms; for closed s, these groups are often torsion-free and computable via . Geometric analysis leverages the compactness of closed manifolds to address variational problems on metrics and submanifolds. The Yamabe problem seeks a conformal metric of constant scalar curvature on a closed (M,g)(M, g), and its positive solution relies on the Sobolev embedding theorems valid due to compactness, ensuring the existence of minimizers in the conformal class. For minimal hypersurfaces, their embedding in closed manifolds of dimension 3 to 7 is guaranteed infinitely often, with area bounds arising from index estimates that exploit compactness to control bubbling phenomena. A fundamental theorem in states that every closed manifold admits a Morse function—a smooth function with non-degenerate critical points—possessing only finitely many critical points, a direct consequence of the ensuring that critical points are isolated and the manifold is covered by finitely many coordinate charts around them. This finiteness underpins handle decompositions and homotopy equivalences in manifold topology.

References

  1. https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
  2. https://www.math.stonybrook.edu/~oleg/courses/mat530.fall19/TopMfds.pdf
  3. https://math.berkeley.edu/~donoghue/reu/quantum/Notes_June_26.pdf
  4. https://math.mit.edu/~mrowka/math965lectnote.pdf
  5. https://ncatlab.org/nlab/show/compact+topological+space
  6. http://math.uchicago.edu/~may/REU2020/AgnesSlides.pdf
  7. https://www.math.stonybrook.edu/~oleg/courses/mat530.fall21/TopMfds.pdf
  8. https://mathworld.wolfram.com/Manifold.html
  9. https://people.math.harvard.edu/~dafr/M392C-2012/Notes/lecture1.pdf
  10. http://www2.math.uni-wuppertal.de/~scholz/preprints/Concept-of-manifold.pdf
  11. https://www.ms.uky.edu/~guillou/F13/551Notes-Week13.pdf
  12. https://ncatlab.org/nlab/show/line+with+two+origins
  13. https://www.nsm.buffalo.edu/~badzioch/MTH427/_static/mth427_notes_21.pdf
  14. https://academicweb.nd.edu/~andyp/notes/ClassificationSurfaces.pdf
  15. https://math.mit.edu/~robinz/files/PoincareDuality.pdf
  16. https://www.math.colostate.edu/~clayton/courses/601/601_11.pdf
  17. https://webhomes.maths.ed.ac.uk/~v1ranick/papers/davenema.pdf
  18. https://mathoverflow.net/questions/58061/how-can-there-be-topological-4-manifolds-with-no-differentiable-structure
  19. https://projecteuclid.org/journals/algebraic-and-geometric-topology/volume-20/issue-7/A-note-on-the-complexity-of-hcobordisms/10.2140/agt.2020.20.3313.pdf
  20. https://www.ime.usp.br/~gorodski/teaching/mat5771/ch1.pdf
  21. https://www.cis.upenn.edu/~cis6100/cis61008Riem-conn.pdf
  22. https://www.maraha.dk/Existence_of_closed_geodesics.pdf
  23. https://link.springer.com/article/10.1007/s12220-025-02076-3
  24. https://academiccommons.columbia.edu/doi/10.7916/rgxw-ve73
  25. https://mathoverflow.net/questions/118429/fundamental-group-of-a-compact-manifold
  26. https://math.stackexchange.com/questions/744824/is-the-fundamental-group-of-a-compact-manifold-finitely-presented
  27. https://link.springer.com/book/10.1007/978-1-4419-7940-7
  28. https://web.math.ucsb.edu/~long/pubpdf/asymmetric.pdf
  29. https://math.mit.edu/~hrm/palestine/lee-smooth-manifolds.pdf
  30. https://www3.nd.edu/~lnicolae/ind-thm.pdf
  31. https://sites.oxy.edu/rnaimi/pastcourses/topology/460f05/classnotes/sec07.pdf
  32. https://mathworld.wolfram.com/CompactManifold.html
  33. https://www.ime.usp.br/~gorodski/teaching/mat5771/ch3.pdf
  34. http://staff.ustc.edu.cn/~wangzuoq/Courses/16S-RiemGeom/Notes/Lec15.pdf
  35. https://www.math.ucla.edu/~petersen/manifolds.pdf
  36. https://www.math.ksu.edu/~sadykov/4-manifolds.pdf
  37. https://www.claymath.org/wp-content/uploads/2022/06/poincare.pdf
  38. https://projecteuclid.org/journals/journal-of-differential-geometry/volume-17/issue-3/The-topology-of-four-dimensional-manifolds/10.4310/jdg/1214437136.pdf
  39. https://arxiv.org/abs/0711.4630
  40. https://arxiv.org/abs/hep-th/9702155
  41. https://arxiv.org/abs/0802.0634
  42. https://doi.org/10.1142/S0218271807009826
  43. https://arxiv.org/abs/2510.13971
  44. https://projecteuclid.org/journals/geometry-and-topology/volume-21/issue-3/Building-Anosov-flows-on-3manifolds/10.2140/gt.2017.21.1837.full
  45. https://www.cambridge.org/core/books/lectures-on-ergodic-theory-and-pesin-theory-on-compact-manifolds/08C07B555D23C89621ED3D58C8E864D0
  46. https://arxiv.org/abs/2410.00560
  47. https://arxiv.org/abs/2011.11811
  48. https://www.math.cmu.edu/~rneumaye/YamabeProblem.pdf
  49. https://annals.math.princeton.edu/2023/197-3/p01
  50. https://web.math.utk.edu/~freire/teaching/m663f21/Morse_Theory_Notes.pdf
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