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Surgery theory

Surgery theory is a collection of techniques in for modifying one finite-dimensional manifold to produce another in a controlled manner, primarily by excising embedded spheres and replacing them with disks of complementary dimensions, enabling the up to equivalence, , or , especially in dimensions five and higher. The theory originated in the 1960s with John Milnor's introduction of on differentiable manifolds to study exotic spheres, and was rapidly extended to piecewise-linear (PL) and topological categories by William Browder, Michel Kervaire, Sergei Novikov, , and C.T.C. Wall, who developed the foundational and obstruction groups. Their work disproved the manifold Hauptvermutung, showing that equivalent manifolds may not be homeomorphic, and built on earlier results like the theorem by . Later contributions by Frank Quinn, Andrew Ranicki, and Wolfgang Lück incorporated algebraic tools, including L-theory and quadratic forms, to handle more general settings such as Poincaré complexes and infinite fundamental groups. At its core, surgery theory revolves around normal maps—degree-one maps from a manifold to a Poincaré complex equipped with stable bundle data—and the surgery obstruction, an algebraic invariant in the L-groups L_n(\mathbb{Z}[\pi_1(X)], w) that determines whether a homotopy equivalence can be realized by . The process involves iterative steps to kill homotopy groups below half the dimension, using tools like the trick for transversality and Spivak normal fibrations to ensure compatibility. The exact sequence relates the structure set S_{TOP}(X), which parametrizes homotopy manifolds over a complex X, to normal invariants in [X, G/TOP] and L-groups, providing a computable framework for obstructions. Applications of surgery theory include the classification of exotic spheres via the stable of spheres, resolutions of the topological rigidity problem, and connections to broader conjectures like the Farrell-Jones and Baum-Connes assemblies in equivariant . It has also influenced the study of manifold structures on types and the enumeration of classes in high dimensions.

Surgery on Manifolds

Fundamental Observation

A manifold is a that is locally , meaning that around every point there exists a neighborhood homeomorphic to an open subset of \mathbb{R}^n. In the context of surgery theory, compact smooth manifolds of dimension n are particularly amenable to into handlebodies or CW-complexes, where the space is constructed iteratively by attaching cells (disks D^k) along their boundaries via maps from spheres S^{k-1}. This handle decomposition provides a framework for understanding the manifold's structure, as handles of various indices capture both the geometric and topological features of the space. The core insight of surgery theory, often termed the fundamental observation, is that in dimensions n \geq 5, a degree-1 normal map from a manifold to a Poincaré complex can be modified through local surgeries—specifically, excising the of an embedded S^k \subset M^n (with k < n/2) and attaching a (k+1)-handle along the boundary S^k \times S^{n-k-1}—to kill non-trivial classes in \pi_k(M) below the middle dimension, yielding a highly connected map while producing an h-cobordism to the original manifold. This process preserves the homotopy type relative to the target complex but alters the diffeomorphism class of the manifold, enabling the classification of manifold structures on homotopy types. This observation is underpinned by the h-cobordism theorem, which equates h-cobordant manifolds (those connected by such handle attachments with trivial Whitehead torsion) with diffeomorphisms relative to the boundary in dimensions at least 5. Central to this observation is the embedding of spheres into the manifold interior, where a smoothly embedded sphere S^k \subset M^n (with k < n/2) admits a tubular neighborhood diffeomorphic to a disk bundle in its normal bundle. In dimensions n \geq 5, the Whitney trick ensures that such embeddings can be made transverse (hence disjoint from other submanifolds of complementary dimension), and the normal bundle can often be trivialized stably, facilitating the handle attachment without affecting compatibility with the ambient structure. The role of the normal bundle is pivotal, as it determines the framing for the attaching map of the replacement handle, ensuring the surgery aligns smoothly with the original manifold. Consequently, in dimension n \geq 5, surgery enables controlled local modifications that relate homotopy equivalent manifolds, allowing algebraic classification of diffeomorphism types via obstructions in L-groups and normal invariants, as surgeries produce h-cobordisms. This fact stems directly from the stable range of embeddings and handle cancellation lemmas, which hold reliably above dimension 4.

Definition of the Surgery Operation

The surgery operation in manifold topology is a precise technique for modifying an n-dimensional manifold M^n by excising a neighborhood of an sphere and reattaching a handle, thereby altering its topological structure in a controlled manner. This operation, introduced as a fundamental tool in the classification of manifolds, relies on the existence of an of a k-dimensional sphere S^k into M^n with a trivial normal bundle. The process begins with selecting a smoothly (or PL) embedded sphere S^k \subset M^n, where $0 \leq k < n/2 typically, ensuring the embedding is framed, meaning the normal bundle \nu(S^k) is trivialized. A tubular neighborhood \nu(S^k) of this sphere, diffeomorphic to S^k \times D^{n-k}, is then removed from its interior, leaving M^n with a boundary component diffeomorphic to S^k \times S^{n-k-1}. Finally, a handle D^{k+1} \times S^{n-k-1} is attached along this boundary via the identity map induced by the framing, yielding the new manifold M' = (M^n - \mathrm{int}(\nu(S^k))) \cup_{S^k \times S^{n-k-1}} (D^{k+1} \times S^{n-k-1}). This construction preserves the orientability and dimension of the original manifold while modifying its homotopy type, particularly by killing the homotopy class represented by the embedded sphere. For the surgery to be well-defined in the smooth or PL category, the normal bundle \nu(S^k) must be trivial, which is guaranteed in high dimensions by the existence of framings for spheres in stable range, and the manifold dimension satisfies n \geq 5 to allow the Whitney trick for disentangling intersections. In codimension n - k \geq 3, the embedding can be chosen without self-intersections, ensuring the resulting manifold M' is smooth (or PL) and unique up to diffeomorphism (or PL homeomorphism), as isotopic embeddings yield isotopic results. This codimension condition prevents singularities in the attachment and enables the operation to be performed without altering the fundamental group or lower-dimensional topology unexpectedly.

Handlebodies and Cobordisms

Handlebodies are compact manifolds constructed by starting with a base manifold with boundary and successively attaching handles along embeddings in its boundary. Formally, an n-dimensional k-handle is the product D^k \times D^{n-k}, attached to the boundary via an embedding of its "attaching region" S^{k-1} \times D^{n-k}. A handlebody W of dimension n with distinguished boundary components \partial_0 W and \partial_1 W is then obtained by taking \partial_0 W \times [0,1] as the initial piece and attaching a finite collection of such k-handles (for $0 \leq k \leq n) along embeddings into the evolving boundary component corresponding to \partial_1 W \times \{1\}. This construction provides a handle decomposition of W relative to \partial_0 W, mirroring the structure arising from on manifolds. In surgery theory, the operation of performing surgery on an embedded k-sphere in an n-manifold M is geometrically realized as attaching a (k+1)-handle to M \times [0,1]. Specifically, if \iota: S^k \times D^{n-k} \hookrightarrow M is an embedding of a tubular neighborhood of the sphere, surgery removes the interior of \iota(D^{k+1} \times D^{n-k-1}) and glues in D^{k+1} \times S^{n-k-1} along the boundary S^k \times S^{n-k-1}, yielding a new manifold M'. This process corresponds precisely to the attachment of a (k+1)-handle, altering the topology of M in a controlled manner to kill elements in homotopy groups. Such handle attachments form the core mechanism of surgery, enabling the modification of manifold structures while preserving certain invariants. A sequence of surgeries on M generates a cobordism W between the original manifold M and the resulting manifold M', where W is the trace of the attachments: W = M \times [0,1] \cup_{\iota_1} (D^{k_1+1} \times D^{n-k_1}) \cup \cdots \cup_{\iota_s} (D^{k_s+1} \times D^{n-k_s}). The boundary of this cobordism satisfies \partial W = -M \sqcup M', establishing W as a compact (n+1)-manifold with the two n-manifolds as its boundary components. These cobordisms provide the framework for relating distinct manifolds topologically, with the handle structure of W encoding the surgical modifications. In the simply-connected case, handle attachments of index below the middle dimension do not alter the homotopy type of the manifold.

Basic Examples

One of the simplest illustrations of the surgery operation occurs in dimension 3, where an embedded circle S^1 (an unknot) in the 3-sphere S^3 serves as the submanifold for excision. Removing a tubular neighborhood of this S^1, which is diffeomorphic to a solid torus S^1 \times D^2, leaves another solid torus as the complement. Gluing back a solid torus via a specific framing—known as 0-surgery, where the meridian of the new solid torus is attached to the longitude of the excised neighborhood—yields the 3-manifold S^1 \times S^2. This operation demonstrates how surgery can alter the fundamental group from trivial (as in S^3) to \mathbb{Z}, while preserving the homology in higher degrees. Performing the same 0-surgery simultaneously on two disjoint unknotted embeddings of S^1 in S^3 produces the connected sum (S^1 \times S^2) \# (S^1 \times S^2), as the surgeries act independently on disjoint neighborhoods. This example highlights the additive nature of surgery under connected sums, transforming the simply connected S^3 into a manifold with fundamental group \mathbb{Z} * \mathbb{Z} and Betti number b_1 = 2. In both cases, the boundary tori from the handle attachments (referencing the cobordism structure) play a key role in the gluing, but the resulting closed manifolds exhibit distinct toroidal factors absent in the original S^3. In contrast, surgery with framing \pm 1 on a non-trivial knot in S^3 often produces a 3-sphere, a manifold with the same as S^3 but potentially non-trivial fundamental group. For instance, +1 surgery on the right-handed trefoil knot—a (2,3)-torus knot—yields the Poincaré sphere, first described by Henri Poincaré in 1904 as the boundary of a dodecahedral orbifold. This manifold has fundamental group the binary icosahedral group of order 120 and is a classic example of a sphere that is not diffeomorphic to S^3, illustrating how surgery can introduce finite non-abelian fundamental groups while preserving integral . Michel Kervaire provided early constructions of such spheres via successive surgeries: starting from a connected sum of copies of S^1 \times S^{n-1} (for n > 4), embedding spheres to kill relators in a , and performing additional 2-surgeries on S^2 \times D^{n-2} to achieve vanishing in dimensions 1 and 2. These examples underscore surgery's role in generating exotic homotopy types from standard manifolds. A notable dimension-specific illustration arises in 4-manifold topology, where along an embedded \mathbb{RP}^2 () in \mathbb{CP}^2 () reveals challenges unique to dimension 4, such as obstructions to smooth structures and alterations to the intersection form. The embedding of \mathbb{RP}^2 in \mathbb{CP}^2 has a twisted , and performing 2-surgery—excising \mathbb{RP}^2 \times D^2 and gluing in D^3 \times S^1—can produce manifolds with altered Kirby-Siebenmann invariants, potentially non-smoothable in the topological category. This operation, akin to twisting along the \mathbb{RP}^2, modifies the second Stiefel-Whitney class and highlights how middle-dimensional in 4-manifolds can lead to exotic phenomena, like failure of the smooth h-cobordism theorem, unlike in higher even dimensions where such issues diminish.

Topological and Homotopy Consequences

Effects on Homotopy and Homology Groups

Surgery on a manifold M^n along an embedded k-sphere, where $0 < k < n-1, primarily affects the groups by killing the homotopy class represented by that sphere in \pi_k(M). This operation attaches a handle D^{k+1} \times D^{n-k} to M, effectively adding relations that eliminate the specified element in \pi_k(M), while the attaching map of the handle influences higher groups through the resulting cell structure changes. In particular, for surgery on an embedding f: S^k \times D^{n-k} \to M, the relative group \pi_{k+1}(M, S^k) determines the extent of these modifications, often reducing the rank or introducing torsions in higher \pi_i(M) for i > k. The \pi_1(M') of the resulting manifold M' is computed using the Seifert-van Kampen theorem applied to the decomposition along the surgery neighborhood, yielding a of the free product \pi_1(M \setminus \nu(S^k)) * \pi_1(S^{n-k-1} \times D^{k+1}) by relations imposed by the meridional loops and the attaching . This typically preserves the surjectivity of the inclusion-induced map \pi_1(M) \to \pi_1(M') when k \geq 2, but can introduce new relations if the is non-trivial in \pi_1(M). For higher groups, the changes propagate via the long exact sequence of the pair (M', M), where surgery obstructions in L-groups detect whether the type is preserved up to equivalence. Regarding homology groups, the Mayer-Vietoris for the M' = (M \setminus \nu(S^k)) \cup (D^{k+1} \times S^{n-k-1}) implies that H_*(M') \cong H_*(M) in most degrees, with potential shifts or modifications occurring specifically in dimensions k+1 and n-k-1. The ensures that the H_{k+1}(M', M; \mathbb{Z}) is generated by the core of the attached , while the component contributes to H_{n-k-1}(M', M; \mathbb{Z}), often resulting in an H_i(M') \cong H_i(M) for i \neq k+1, n-k-1 and exact sequences relating the differing degrees via the of the sphere and disk. In the stable range where k < n/2, surgery below the middle dimension preserves the homology groups entirely, as the attachments lie outside the range affecting the manifold's overall Betti numbers, though homotopy alterations may still occur trivially without changing the homotopy type.

Comparison to CW-Cell Attachments

In CW-complex theory, attaching an n-cell to a space X involves specifying a map from the boundary sphere S^{n-1} to X and gluing the disk D^n along this attaching map, which effectively kills the homotopy class represented by that map in \pi_{n-1}(X), thereby modifying the homotopy type of the resulting complex. This process builds CW-complexes inductively, preserving cellular structure while altering fundamental groups and higher homotopy groups in a controlled algebraic manner. Surgery on manifolds provides a smooth analog to this cell attachment mechanism, where an n-surgery on an m-manifold M (with m > n + 1) removes the interior of an embedded disk bundle S^{n-1} \times D^{m-n+1} and attaches a D^n \times D^{m-n+1} along the boundary, mimicking the attachment of an n- while ensuring the result remains a manifold. The homotopy effect parallels that of CW-attachment: the surgery kills the class in \pi_{n-1}(M) represented by the attaching sphere, quotienting the accordingly, much like the reduction observed in cell attachments. This analogy extends to the dual operation, where surgery detaches a complementary to restore , unlike pure CW-attachments which do not inherently preserve such geometric duality. Despite these similarities in homotopy modification—both processes aim to achieve homotopy equivalences by targeted alterations—the frameworks differ fundamentally in their categorical settings. CW-cell attachments operate in the combinatorial or piecewise-linear (PL) category, applicable to general topological spaces without embedding obstructions, whereas surgery resides in the smooth (differentiable) category for manifolds, necessitating the existence of smoothly embedded spheres with trivial normal bundles to ensure the handle attachment yields a smooth manifold. In low dimensions, smooth embeddings may fail due to knotting phenomena, but the Haefliger-Weber theorem guarantees that for embeddings of k-spheres in \mathbb{R}^m with 2m ≥ 3k + 3 (the metastable range), such embeddings exist up to isotopy, enabling surgery in sufficiently high dimensions without smoothing obstructions that plague CW methods in the smooth realm. J.H.C. 's development of simple theory in the bridges these approaches, establishing that h-cobordisms—cobordisms where the inclusions of the boundary manifolds induce s—can be analyzed via Whitehead torsion, a measure of deviation from simple that links manifold to CW-complex deformations. This connection underscores how refines CW-attachment ideas geometrically, with trivial torsion implying the cobordism is diffeomorphic to a product, thus preserving the type across categories.

Classification of Manifolds

Overview of the Surgery Program

The surgery program in seeks to classify or topological manifolds up to equivalence or by systematically modifying a given equivalence f: M \to N between closed n-manifolds into a diffeomorphism or homeomorphism through controlled geometric alterations. The core strategy begins by approximating f with a normal map, which is a degree-one equipped with a bundle from the stable normal bundle of M to that of N, ensuring the respects the embedding in up to . Subsequent steps involve performing surgeries—precise excisions and gluings of handlebodies along embedded spheres—to successively kill elements in the relative groups \pi_i(f) for i < n/2, reducing obstructions until the becomes a homotopy equivalence between simply connected manifolds. Finally, the h-cobordism theorem (for topological category) or s-cobordism theorem (for smooth category) is invoked to confirm whether the resulting equivalence is isotopic to a diffeomorphism or homeomorphism, provided no further obstructions arise. This program was initiated by John Milnor in his 1961 paper, where he introduced the basic surgery technique to kill homotopy groups in differentiable manifolds, laying the groundwork for altering manifold structures while preserving homology. It was rapidly developed in the 1960s by Michel Kervaire, C.T.C. Wall, and Dennis Sullivan, who extended the method to classify simply connected manifolds in high dimensions, incorporating algebraic invariants to measure the success of these modifications. Their contributions, building on Milnor's ideas, established surgery as a primary tool for resolving the Hauptvermutung and related classification problems in dimensions greater than or equal to 5. The program succeeds in dimensions n \geq 5, where the stable range for homotopy groups and the h/s-cobordism theorems apply effectively, allowing complete classification up to diffeomorphism or homeomorphism for simply connected cases. However, in dimension 4, significant challenges persist due to the existence of exotic smooth structures on manifolds like \mathbb{CP}^2 \# \overline{\mathbb{CP}^2}, as demonstrated by Simon Donaldson's gauge-theoretic constructions in the 1980s, which reveal that smooth and topological categories diverge dramatically. These exotic phenomena, arising from Yang-Mills invariants, indicate that the surgery program remains incomplete for 4-manifolds, motivating ongoing research into gauge theory and symplectic geometry.

Structure Sets

In surgery theory, structure sets serve as algebraic invariants that measure the difference between homotopy equivalence classes and diffeomorphism (or homeomorphism) classes of manifolds over a given Poincaré complex. For a Poincaré complex X of dimension n, the structure set S(X) is defined as the set of homotopy equivalence classes of maps f: M \to X, where M is a closed n-manifold and f is a degree-one homotopy equivalence, taken modulo homotopy equivalences relative to X. Equivalently, it consists of bordism classes of normal maps (M, \nu_M, f: M \to X) with stable trivial normal bundle, where two such maps are equivalent if they bound an (n+1)-dimensional h-cobordism over X. Variants of the structure set arise depending on the category of manifolds considered. The topological structure set S^{\text{TOP}}(X) classifies topological manifolds up to homeomorphism, the smooth structure set S^{\text{DIFF}}(X) classifies smooth manifolds up to diffeomorphism, and the piecewise-linear structure set S^{\text{PL}}(X) classifies PL manifolds up to PL homeomorphism. These sets often relate to classifying spaces via S^{\text{DIFF}}(X) \cong [X, G/O], S^{\text{PL}}(X) \cong [X, G/\text{PL}], and S^{\text{TOP}}(X) \cong [X, G/\text{TOP}], where G is the infinite special orthogonal group and the quotients encode stable tangential structures. The differences between these variants capture obstructions to triangulations or smoothings, such as the exotic smooth structures on topological manifolds. Computations of structure sets rely on their relation to algebraic invariants, particularly the quadratic L-groups L_n(\mathbb{Z}[\pi_1(X)]) and Witt groups of quadratic forms over the group ring \mathbb{Z}[\pi_1(X)]. The L-groups, which are 4-periodic and detect surgery obstructions via signatures and Arf invariants, appear in exact sequences mapping onto or acting on S(X); for instance, elements of L_{n+1}(\mathbb{Z}[\pi_1(X)]) act on S(X) through Wall realization, realizing quadratic forms as differences between manifold structures. Witt groups, isomorphic to even-dimensional L-groups L_{2k}, classify stable isomorphism classes of non-singular quadratic forms and provide the algebraic framework for these obstructions. A prominent example occurs for spherical Poincaré complexes X = S^n. In the topological and PL categories, |S(S^n)| = 1 for all n \geq 1, reflecting the uniqueness of the standard sphere up to homeomorphism or PL equivalence by the Poincaré conjectures in these categories across all dimensions. However, in the category, non-trivial structure sets arise from exotic spheres: Milnor demonstrated the existence of exotic 7-spheres, smooth manifolds homeomorphic but not diffeomorphic to S^7, showing |S^{\text{DIFF}}(S^7)| > 1. Kervaire and Milnor later computed that there are exactly 28 smooth structures on the 7-sphere, forming the \Theta_7 \cong \mathbb{Z}/28\mathbb{Z}, with similar but trivial results for most other dimensions where |\Theta_n| = 1 except in specific cases like n=7,15.

Surgery Exact Sequence

The surgery exact sequence serves as the algebraic core of surgery theory, providing a long exact sequence that relates the structure set of a manifold to algebraic invariants derived from quadratic forms and homotopy data. For a closed oriented n-manifold M with fundamental group π₁, the sequence takes the form \dots \to L_n(\mathbb{Z}[\pi_1]) \to S_n(M) \to [M, G/\mathrm{TOP}] \to L_{n-1}(\mathbb{Z}[\pi_1]) \to \dots, where the L_k(ℤ[π₁]) denote Wall's surgery obstruction groups, which classify quadratic forms up to cobordism over the group ring ℤ[π₁]. These groups capture obstructions to surgery on normal maps, with even-dimensional L_{2k} groups arising from metabolic Hermitian forms and odd-dimensional L_{2k+1} from their automorphism groups. The term S_n(M) is the n-dimensional topological structure set, consisting of homotopy classes of orientation-preserving homotopy equivalences from compact oriented n-manifolds to M, modulo h-cobordisms. The space [M, G/TOP] represents homotopy classes of maps from M to the classifying space for stable spherical fibrations, encoding normal invariants that specify stable normal bundle data over M. A degree-one normal map f: N → X, where X is a Poincaré complex homotopy equivalent to M and N is an n-manifold, induces a normal invariant η(f) ∈ [X, G/TOP] that measures the difference between the stable normal bundle of N and a bundle pulled back from X. The connecting homomorphism ∂: [M, G/TOP] → L_{n-1}(ℤ[π₁]) assigns to η(f) the primary surgery obstruction, which lies in the kernel of the forgetful map from the group of normal bordisms N_n(X) to the L-group; this obstruction vanishes if and only if f is normally bordant to a homotopy equivalence. Conversely, the map σ: S_n(M) → [M, G/TOP] sends a homotopy equivalence h: M' → M to the normal invariant of the associated normal map, reflecting how manifold structures differ from the given homotopy type up to stable bundle data. The surgery kernel, comprising the homotopy fiber of f over X, contributes to secondary obstructions in higher L-groups via the boundary map δ: L_n(ℤ[π₁]) → S_n(M). The derivation of the sequence proceeds from geometric considerations of normal bordism and to an algebraic framework, where surgery obstructions on the geometric level—such as Arf invariants detecting non-metabolic forms in simply connected cases—are realized via algebraic Hermitian and forms over ℤ[π₁]. These forms encode intersection data from the kernels of maps, with exactness arising from the Rothenberg sequence relating projective and simple L-groups and the assembly map from - and L-homology to algebraic L-groups, without requiring a full proof here. Exactness of the sequence implies that the map S_n(M) → [M, G/TOP] is surjective whenever the boundary homomorphism [M, G/TOP] → L_{n-1}(ℤ[π₁]) vanishes, meaning every normal invariant lifts to a manifold structure on M if the corresponding surgery obstruction is zero; this key surjectivity result underpins the in simply equivalent classes and was established by and in the .

Key Theorems and Obstructions

The h-cobordism theorem, established by in 1962, asserts that if two simply-connected smooth closed n-manifolds with n ≥ 5 are h-cobordant via a compact smooth , then the cobordism is diffeomorphic to the product of one manifold with the [0,1]. This result forms a cornerstone of high-dimensional manifold classification, implying the in dimensions greater than four. The s-cobordism theorem, developed by Dennis Barden, , and John Stallings around 1964-1965, extends the theorem to non-simply-connected cases by incorporating Whitehead torsion invariants; specifically, an s-cobordism between manifolds of dimension at least 6 is a product if and only if the torsion vanishes. This generalization, building on the framework, enables the study of equivalences up to in broader settings. In surgery theory, primary obstructions to realizing homotopy equivalences as diffeomorphisms lie in the algebraic L-groups L_n(ℤ[π_1(M)]), where for even dimensions they are captured by the and for odd dimensions by the Arf invariant. These invariants, rooted in forms over rings with , detect whether a normal map can be surgically modified to a equivalence. For non-simply-connected manifolds, secondary obstructions arise in the algebraic K-group SK_1(ℤ[π_1(M)]), measuring deviations from stable isomorphism in the ring. Surgery theory remains incomplete in dimension 4, where Freedman's 1982 classification of topological 4-manifolds by their intersection form and Kirby-Siebenmann succeeds, but counterparts fail due to exotic phenomena. As of 2025, questions about exotic structures on ℝ^4 persist, with infinitely many known but unresolved issues regarding their types and relations to the Poincaré conjecture. Rokhlin's theorem from 1952 demonstrates that no closed has congruent to 8 16, implying divisible by 16 and obstructing certain that would produce such manifolds. This invariant provides a fundamental barrier in low-dimensional , linking topological and analytic constraints.