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Local system

In mathematics, particularly in algebraic topology and sheaf theory, a local system on a topological space X is defined as a sheaf \mathcal{F} of abelian groups (or more generally, of modules or vector spaces over a ring) such that for every point x \in X, there exists a neighborhood U of x on which the restriction \mathcal{F}|_U is a constant sheaf.^[1] This structure allows the sheaf to vary in a controlled, "twisted" manner across X, reflecting the topology while remaining constant locally.^[2] Local systems were introduced by Norman Steenrod in 1943 as a framework for defining homology with local coefficients, enabling the computation of topological invariants that account for non-trivial monodromy around loops in the space.^[3] Equivalently, a local system on X corresponds to a representation of the fundamental groupoid of X (or, when based at a point x_0, of the fundamental group \pi_1(X, x_0)) into the automorphism group of the stalk of the sheaf, providing a functorial link between algebraic representations and geometric data.^[2] This equivalence arises from the fact that the monodromy action along paths in X determines the transition functions of the sheaf, and vice versa.^[1] Local systems play a central role in twisted cohomology theories, where cohomology groups with coefficients in a local system \mathcal{F} capture invariants sensitive to the space's fundamental group, generalizing ordinary sheaf cohomology.^[2] For instance, on simply connected spaces, every local system is constant, reducing to standard coefficients, but on spaces like the punctured plane \mathbb{C} \setminus \{0\}, non-trivial examples such as the square-root sheaf illustrate how local systems encode branching phenomena.^[1] They also relate to flat connections on vector bundles via the Riemann-Hilbert correspondence, bridging differential geometry and topology.^[2]

Definition

Locally Constant Sheaves

In algebraic topology, a sheaf on a topological space X is a contravariant functor from the poset of open subsets of X to the category of abelian groups (or modules over a ring R) that satisfies two key axioms: the identity axiom, ensuring that a section uniquely determined locally is globally unique, and the gluing axiom, allowing compatible local sections over a cover to be glued into a global section over the union. This structure captures local data on X that can be assembled compatibly, with the stalk \mathcal{F}_x at a point x \in X defined as the direct limit of sections over neighborhoods of x, representing the "germ" of sections at x.^[4] A local system \mathcal{L} on a topological space X is a sheaf of abelian groups (or R-modules) that is locally constant, meaning that for every point x \in X, there exists an open neighborhood U of x such that the restriction \mathcal{L}|_U is isomorphic to the constant sheaf \underline{\mathcal{L}_x} associated to its stalk \mathcal{L}_x. The constant sheaf \underline{A} for a fixed abelian group A assigns to each open set V the group A^{\pi_0(V)} of locally constant functions from the connected components of V to A, with restrictions preserving these components. This local constancy ensures that \mathcal{L} varies "constantly" in small neighborhoods, making it suitable for defining coefficients in cohomology theories that account for topological twisting.^[5]^[6] The concept of local systems originated with Norman Steenrod's work on homology with local coefficients, where they served as a framework for handling varying coefficient groups over a space, predating the formal development of sheaf theory. In Steenrod's formulation, these systems allowed homology computations to incorporate local variations tied to the topology of the base space, such as in fiber bundles or covering spaces.^[3] A key property of local systems is that their stalks \mathcal{L}_x are typically finite-dimensional vector spaces over a field k (such as \mathbb{C}), ensuring finite rank and enabling analytic or geometric interpretations. The transition functions between local trivializations on overlapping open sets U_i \cap U_j are constant on each connected component of the intersection, reflecting the sheaf's local constancy and guaranteeing compatibility across the space without introducing unnecessary variation. This structure distinguishes local systems from more general sheaves, like coherent sheaves in algebraic geometry, by emphasizing topological rather than analytic or algebraic constraints.^[5]^[7]

Monodromy Representation

A local system \mathcal{L} on a pointed topological space (X, x_0) is equivalently defined by a representation \rho: \pi_1(X, x_0) \to \Aut(\mathcal{L}_{x_0}), where \mathcal{L}_{x_0} denotes the stalk (fiber) at the basepoint x_0 and \Aut(\mathcal{L}_{x_0}) is the group of automorphisms of this fiber, typically a vector space or module over a ring.^[7]^[5] This representation captures the parallel transport of sections along loops based at x_0, assigning to each homotopy class [\gamma] \in \pi_1(X, x_0) an automorphism \rho([\gamma]) that describes how the fiber twists when transported around \gamma.^[8] The equivalence arises because the local system's structure sheaf allows consistent identification of fibers over contractible neighborhoods, enabling the global action of the fundamental group on the base fiber to define the entire sheaf.^[7] The monodromy action is explicitly given by \rho(\gamma) \cdot v for a loop \gamma \in \pi_1(X, x_0) and v \in \mathcal{L}_{x_0}, where \rho(\gamma) is the automorphism induced by lifting \gamma to a path in the total space of the associated étale space and transporting v along this lift.^[7] This action extends to the whole sheaf \mathcal{L} by path lifting: for any path \alpha: [0,1] \to X with \alpha(0) = x_0, parallel transport defines an isomorphism \mathcal{L}_{x_0} \to \mathcal{L}_{\alpha(1)} compatible with homotopy, ensuring that the representation \rho reconstructs \mathcal{L} as the sheaf of locally constant sections over X.^[5] If two paths \alpha, \beta are homotopic relative to endpoints, their induced isomorphisms coincide, making the monodromy well-defined up to homotopy.^[8] The local constancy of \mathcal{L} implies that the representation \rho is continuous when \Aut(\mathcal{L}_{x_0}) is equipped with the discrete topology, as fibers over simply connected open sets are canonically identified without twisting, and the action only varies globally via the fundamental group.^[7] In this topology, every map from the discrete space \pi_1(X, x_0) is continuous, aligning the topological and algebraic structures seamlessly.^[5] This discreteness ensures that the sheaf is étale over X, with the total space being a covering space modulo the group action.^[7] Unlike constant sheaves, where the representation \rho is trivial (i.e., \rho(\gamma) = \mathrm{id} for all \gamma), local systems permit non-trivial twisting, allowing the fiber to vary systematically under the fundamental group's action and capturing phenomena like orientation reversals or more complex bundle structures.^[8] This distinction enables local systems to model local coefficients in homology and cohomology, generalizing constant coefficient theories to spaces with non-trivial topology.^[5]

Formulations and Spaces

Path-Connected Spaces

In the case of a path-connected topological space X, the definition of a local system simplifies significantly compared to the general setting. Specifically, every local system on X with fiber V (a module over a commutative ring R) is determined up to isomorphism by its monodromy representation \rho: \pi_1(X, x_0) \to \Aut_R(V), where x_0 \in X is a basepoint and \Aut_R(V) denotes the group of R-linear automorphisms of V. This representation arises from parallel transport along loops based at x_0, and it is independent of the choice of basepoint up to conjugation in \Aut_R(V), due to the path-connectedness of X allowing conjugation by paths between basepoints.^[9]^[10] A fundamental structural result is that the category of local systems on a path-connected space X is equivalent to the category of representations of the fundamental group \pi_1(X) on R-modules. Under this equivalence, the forgetful functor sending a local system to its stalk (fiber) at a basepoint corresponds to the fiber functor on representations, which evaluates the module at the basepoint. This category equivalence holds because the monodromy action fully encodes the gluing data for the locally constant sheaf, and path-connectedness ensures a single representation suffices without additional compatibility conditions across components.^[11]^[12] Local systems also admit geometric interpretations in terms of fiber bundles and connections. For local systems of sets (i.e., R = \mathbb{Z} and V a discrete set), they correspond precisely to fiber bundles over X with discrete fibers, which are étale covers equipped with a transitive action of the deck transformation group on the fiber. In the vector space case (e.g., R = k a field and V a finite-dimensional k-vector space), local systems are equivalent to vector bundles over X equipped with a flat connection, meaning a connection whose curvature vanishes, ensuring local triviality via parallel transport and compatibility with the monodromy representation. This flatness guarantees that the bundle is locally isomorphic to the trivial bundle X \times V with the trivial connection.^[13]^[14]^[15]

Non-Path-Connected Spaces

In a general topological space X that is not path-connected, a local system \mathcal{L} is defined componentwise on the path components of X. Specifically, if \{X_i\}_{i \in I} denotes the collection of path components of X, then \mathcal{L} consists of a local system \mathcal{L}_i on each X_i, with no required compatibility conditions between the different \mathcal{L}_i. This extends the notion from path-connected spaces, where a single monodromy representation of the fundamental group suffices, to the disjoint union structure inherent in non-path-connected settings.^[16] More precisely, a local system on X corresponds to a functor from the fundamental groupoid \pi_1(X) of X—whose objects are points of X and morphisms are homotopy classes of paths—to the category of vector spaces (or modules over a ring), assigning to each path component an independent representation of its own fundamental group. Since the fundamental groupoid has no morphisms between distinct path components, the local systems on different X_i are independent, allowing potentially different ranks or structures on each component.^[16] A key consequence of this componentwise definition is that the space of global sections decomposes as a product: \Gamma(X, \mathcal{L}) = \prod_{i \in I} \Gamma(X_i, \mathcal{L}_i). This reflects the disjoint nature of the path components, where sections on X are precisely the tuples of sections restricted to each X_i.^[16]

Examples

Trivial and Constant Systems

The trivial local system on a topological space X with values in an abelian group A is the constant sheaf \underline{A} whose sections over any open set U \subseteq X consist of constant functions U \to A, equipped with the trivial monodromy representation \rho: \pi_1(X, x_0) \to \Aut(A) that acts as the identity on A.^[17] This sheaf is globally constant, meaning it is isomorphic to the product bundle X \times A, and its stalks are canonically identified with A at every point.^[17] A key property of the constant sheaf \underline{A} is that its global sections over a connected open set U are precisely the constant functions to A, reflecting the absence of twisting by the fundamental group.^[18] When A is equipped with the discrete topology, \underline{A} coincides with the sheaf of locally constant functions U \to A, ensuring that local systems with discrete stalks capture untwisted coefficient systems in algebraic topology.^[18] For the case where A is a vector space V over a field k, the trivial local system has constant rank equal to \dim_k V across all points of X, as the fiber over each point is isomorphic to V.^[17] Local systems are defined to have discrete stalks, and those with finite stalks (e.g., when A is a finite abelian group) are inherently locally constant, providing the simplest examples without monodromic variation.^[7]

Non-Trivial Geometric Examples

One prominent non-trivial geometric example of a local system arises from the orientation sheaf on a non-orientable manifold such as the real projective plane \mathbb{RP}^2. Here, the sheaf \mathcal{L} has stalks isomorphic to \mathbb{Z} at each point, but the monodromy representation \rho: \pi_1(\mathbb{RP}^2) \to \mathrm{Aut}(\mathbb{Z}) is non-trivial, with \pi_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z} acting via the sign homomorphism: the identity element acts as multiplication by +1, while the generator (corresponding to an orientation-reversing loop) acts by multiplication by -1.^[19]^[20] This twisting reflects the global non-orientability of \mathbb{RP}^2, where local orientations cannot be consistently glued, resulting in a locally constant sheaf that captures sign changes along certain paths.^[19] Another illustrative example is the local system on the punctured plane \mathbb{R}^2 \setminus \{0\}, where the fundamental group \pi_1(\mathbb{R}^2 \setminus \{0\}) \cong \mathbb{Z} is generated by a loop circling the origin once. A non-trivial representation \rho: \mathbb{Z} \to \{\pm 1\} \subset \mathrm{[GL](/page/GL)}(1, \mathbb{R}) sends the generator to -1, inducing a Möbius-like twisting in the sheaf sections: parallel transport around the origin reverses the sign of vectors in the \mathbb{R}-stalks.^[19]^[17] This construction demonstrates how the puncture introduces monodromy that prevents global triviality, even though the space is homotopy equivalent to the circle.^[19] Covering spaces provide a broad class of non-trivial local systems, particularly through the associated sheaves of sets. For an n-sheeted connected covering p: Y \to X, the sheaf \mathcal{L} has stalks of cardinality n, and the monodromy action of \pi_1(X, x_0) on the fiber p^{-1}(x_0) is transitive and free, corresponding to the deck transformation group isomorphic to the subgroup p_*(\pi_1(Y, y_0)).^[19] More generally, vector bundles with flat connections yield local systems of vector spaces, where the representation \rho: \pi_1(X) \to \mathrm{GL}(V) encodes parallel transport along loops.^[19]^[17] Local systems classify flat vector bundles up to isomorphism: there is an equivalence of categories between flat \mathbb{R}-vector bundles on a manifold X (equipped with a connection of vanishing curvature) and local systems of \mathbb{R}-vector spaces on X, via the monodromy representation that identifies parallel transport isomorphisms between fibers.^[21]^[19]

Cohomology

Sheaf Cohomology

Sheaf cohomology with coefficients in a local system \mathcal{L} on a topological space X is defined as the j-th right derived functor of the global sections functor \Gamma(X, -) applied to \mathcal{L}, denoted H^j(X, \mathcal{L}).^[22] Local systems, being locally constant sheaves of abelian groups (or vector spaces), capture twisted coefficients arising from the fundamental groupoid of X. These groups serve as primary invariants measuring the extent to which \mathcal{L} fails to be acyclic globally. One standard method to compute H^j(X, \mathcal{L}) employs the Čech complex associated to an open cover \mathcal{U} of X where \mathcal{L} is constant on the intersections U_{\alpha_0 \dots \alpha_k}, such as a good cover with simply connected finite intersections. In this case, Leray's theorem ensures that the Čech cohomology \check{H}^j(\mathcal{U}, \mathcal{L}) is isomorphic to H^j(X, \mathcal{L}), obtained as the cohomology of the cochain complex of global sections over the nerves of the cover.^[22] Alternatively, for a more sheaf-theoretic approach, an injective resolution $0 \to \mathcal{L} \to I^0 \to I^1 \to \cdots of \mathcal{L} by injective sheaves (often flasque sheaves) yields H^j(X, \mathcal{L}) \cong H^j(\Gamma(X, I^\bullet)), the cohomology of the complex of global sections.^[22] On paracompact manifolds, resolutions by fine sheaves—such as those built from smooth functions using partitions of unity—simplify this, as fine sheaves are acyclic for \Gamma(X, -), meaning H^j(X, F) = 0 for j > 0 when F is fine.^[19] Key properties include the exactness of the global sections functor on acyclic covers: if a cover \mathcal{U} is such that H^k(U_{\alpha_0 \dots \alpha_k}, \mathcal{L}) = 0 for k > 0, then the higher direct images vanish, allowing the Čech complex to faithfully compute the derived functors.^[22] For local systems of finite-dimensional vector spaces over a field on compact manifolds, the cohomology groups H^j(X, \mathcal{L}) are finite-dimensional, reflecting the bounded complexity of the twisting by the fundamental group.^[19] In the derived category of sheaves D^b(X), local systems appear as bounded complexes with cohomology concentrated in degree zero, enabling hypercohomology computations via spectral sequences without delving into full derived functor machinery.^[22] For instance, the orientation sheaf on non-orientable manifolds like the Möbius strip yields non-vanishing H^1 reflecting the sign ambiguity in local trivializations.^[7]

Singular Cohomology Equivalence

Singular cohomology with local coefficients in a local system ℒ on a topological space X, denoted H^*_{\sing}(X; \mathcal{L}), is computed from the cohomology of the twisted singular cochain complex C^*(X; \mathcal{L}). This complex arises from the action of the fundamental groupoid of X on the fibers of \mathcal{L}, incorporating monodromy to account for the variation of coefficients along paths. One standard construction uses the universal cover \tilde{X} of a path-component of X, where the cochain groups are C^n(X; \mathcal{L}) = \Hom_{\mathbb{Z}[\pi_1]}(C_n(\tilde{X}), M), where M is the \mathbb{Z}[\pi_1]-module given by the stalk of \mathcal{L} at a basepoint with \pi_1 = \pi_1(X) acting via deck transformations on \tilde{X} and via the monodromy representation on M. The differential \delta: C^n(X; \mathcal{L}) \to C^{n+1}(X; \mathcal{L}) is induced by the singular boundary map on \tilde{X}, respecting the equivariant structure.^[19] In the simplicial formulation, cochains are functions f assigning to each oriented singular n-simplex \sigma: \Delta^n \to X an element f(\sigma) \in \mathcal{L}_{x_0(\sigma)}, where x_0(\sigma) is the image of the starting vertex under \sigma, up to identification via parallel transport. The coboundary operator is given by

\delta f(\sigma) = \sum_{i=0}^{n+1} (-1)^i \cdot \gamma(\sigma, i) \cdot f(\sigma|_{\partial_i}),

where \gamma(\sigma, i) denotes the monodromy automorphism induced by parallel transport along the path in \sigma from the starting vertex of \sigma to that of the i-th face \sigma|_{\partial_i}, ensuring consistency for non-constant paths. For constant local systems, this reduces to the untwisted singular coboundary. This twisting captures the non-trivial action of loops on coefficients, distinguishing local systems from constant ones.^[23]^[19] Under suitable topological assumptions on X, sheaf cohomology with coefficients in \mathcal{L} is isomorphic to this singular cohomology: for paracompact Hausdorff spaces X that are locally contractible, there is a natural isomorphism H^j(X, \mathcal{L}) \cong H^j_{\sing}(X; \mathcal{L}) for all j \geq 0. This equivalence extends to relative and pair versions (X, A). The proof relies on the Leray theorem, which equates sheaf cohomology to Čech cohomology over fine acyclic covers; on such spaces, Čech cohomology with local coefficients matches singular cohomology via subdivision and homotopy invariance arguments, as the local contractibility ensures simplicial approximations align with sheaf resolutions. Milder conditions, such as semi-locally contractible and paracompact, suffice for the isomorphism when \mathcal{L} is a sheaf of abelian groups. These assumptions ensure the necessary refinements of covers and acyclicity of nerves, bridging the intrinsic sheaf perspective with the combinatorial singular chains.^[19]

Generalizations

Constructible Sheaves

Constructible sheaves generalize local systems by allowing the sheaf to vary across a stratified decomposition of the space, while remaining locally constant within each stratum. Specifically, given a topological space X and a stratification \mathcal{S} = \{S_i\}_{i \in I} consisting of disjoint locally closed subsets whose union is X, a sheaf \mathcal{F} of vector spaces over a field k on X is constructible with respect to \mathcal{S} if its restriction \mathcal{F}|_{S_i} is a locally constant sheaf (i.e., a local system) on each stratum S_i, the stalks \mathcal{F}_x are finite-dimensional k-vector spaces for all x \in X. This framework extends the notion of local constancy from the entire space to piecewise-constant behavior adapted to the geometry of singularities or decompositions.^[24] Local systems fit naturally into this picture as a special case of constructible sheaves. A pure local system on X, being locally constant across the whole space, corresponds to a constructible sheaf with respect to the trivial stratification \mathcal{S} = \{X\}, where the single stratum is X itself and no further decomposition is needed. This inclusion highlights how constructible sheaves capture more general coefficient systems that are constant on strata but may exhibit jumps or monodromy across boundaries, useful for studying spaces with non-trivial topology or singularities.^[25] A foundational result in the theory is the Beilinson-Bernstein-Deligne theorem, which establishes that the category of constructible sheaves on X is an abelian category, and the bounded derived category of constructible sheaves, denoted D^b_c(X, k), admits a t-structure whose heart is the abelian category of perverse sheaves. This t-structure is defined via conditions on the cohomology sheaves' supports relative to the dimension of strata, enabling powerful tools like Verdier duality and the six functor formalism for constructible objects. The theorem provides the categorical foundation for many applications in geometry and representation theory. As an illustrative example, skyscraper sheaves exemplify degenerate constructible sheaves. For a closed point x \in X, the skyscraper sheaf \mathcal{F}_x with stalk k at x and zero elsewhere is constructible with respect to a stratification where \{x\} is one stratum (on which it is locally constant) and X \setminus \{x\} is another (on which it vanishes). This construction demonstrates how constructible sheaves can model Dirac delta-like supports, essential for intersection cohomology and other singular theories.^[24]

Higher Local Systems

Higher local systems extend the classical notion of local systems to higher categorical frameworks, particularly within ∞-category theory. In this context, a higher local system on a space X is defined as a locally constant functor from X to the ∞-category of (∞,n)-categories, or more precisely, as an object in the (n+1)-category (n+1)\mathrm{LocSysCat}_n(X; \mathcal{A}) of n-categorical local systems valued in a presentably symmetric monoidal (∞,n)-category \mathcal{A}.^[26] These structures can be interpreted as representations of the fundamental ∞-groupoid \Pi_\infty(X) of X into an (∞,1)-topos, generalizing the action of the fundamental groupoid on vector spaces in the classical case.^[2] Such representations are equipped with a flat ∞-connection, enabling parallel transport along paths in higher dimensions.^[26] A key advancement in this area is the categorified monodromy equivalence, which describes higher local systems via higher monodromy data. For an (n+1)-connected space X, higher local systems are equivalent to E_{n+1}-modules over the (n+1)-fold based loop space \Omega^{n+1}_* X, generalizing the classical monodromy representation as modules over the based loop space \Omega_* X.^[26] This equivalence extends Teleman's theory of topological actions on categories to (∞,n)-categories and connects invertible higher local systems over an n-connected X to characters of the homotopy group \pi_n(X).^[26] The framework links to stable homotopy theory when \mathcal{A} is the ∞-category of spectra, yielding modules in stable settings, and to derived algebraic geometry through connections with étale cohomology and Brauer groups.^[26] Monodromy in higher dimensions is realized through parallel transport in ∞-bundles, where the structure group acts via E_k-algebra morphisms for k \geq 2.^[26] Examples in loop spaces, such as local systems on simply connected spaces as modules over the double loop space \Omega^2_* X, highlight how these constructions encode higher homotopy information via iterated looping.^[26] This higher-dimensional perspective recovers classical local systems in dimension 1, where n=0 and the E_1-module structure reduces to ordinary representations of the fundamental group.^[26]

Applications

Manifold Duality

In manifold topology, local systems play a crucial role in extending classical duality theorems to incorporate twisted coefficients, allowing for the study of manifolds with non-trivial fundamental group actions on coefficients. The twisted Poincaré duality theorem provides an isomorphism between cohomology and homology groups equipped with a local system \mathcal{L}. For a closed, orientable n-manifold M and a local system \mathcal{L} of rank k vector spaces over a field (or more generally, a \mathbb{Z}\pi_1(M)-module), the theorem states that H^j(M; \mathcal{L}) \cong H_{n-j}(M; \mathcal{L}^\vee \otimes \mathrm{or}_M), where \mathcal{L}^\vee is the dual local system and \mathrm{or}_M is the trivial orientation sheaf (isomorphic to the constant sheaf \underline{\mathbb{Z}} for orientable M).^[19]^[27] This isomorphism is established via the cap product with the fundamental class [M] \in H_n(M; \mathrm{or}_M), yielding a map H^j(M; \mathcal{L}) \to H_{n-j}(M; \mathcal{L}^\vee \otimes \mathrm{or}_M) that is an isomorphism under the given conditions. One proof proceeds by excising submanifolds and applying the Thom isomorphism theorem to their normal bundles: for a codimension-q submanifold N \subset M, a tubular neighborhood U of N retracts to the zero section, and the Thom class in H^q(U, U - N; \mathcal{L}) induces an isomorphism H^j(M - N; \mathcal{L}) \cong H^{j+q}(M, M - N; \mathcal{L}) via the restriction to the boundary of the tubular neighborhood, which extends globally to the duality map.^[19] For non-orientable manifolds, the theorem generalizes by replacing the constant orientation sheaf with the orientation local system \mathrm{or}_M = \mathbb{Z}_w, where w \in H^1(M; \mathbb{Z}/2) is the first Stiefel-Whitney class, yielding H^j(M; \mathcal{L}) \cong H_{n-j}(M; \mathcal{L}^\vee \otimes \mathbb{Z}_w), with the fundamental class now in H_n(M; \mathbb{Z}_w).^[27] A key application arises in computing twisted cohomology groups for non-orientable manifolds like the real projective space \mathbb{RP}^n or the Klein bottle. For \mathbb{RP}^2 with the orientation local system \mathbb{Z}_w, the twisted cohomology is H^0(\mathbb{RP}^2; \mathbb{Z}_w) \cong \mathbb{Z}/2, H^1(\mathbb{RP}^2; \mathbb{Z}_w) = 0, and H^2(\mathbb{RP}^2; \mathbb{Z}_w) \cong \mathbb{Z}, which duality relates to the homology groups via the isomorphism above. Similarly, for the Klein bottle K, whose fundamental group is \langle a, b \mid aba^{-1}b = 1 \rangle, local systems classified by representations \pi_1(K) \to \mathrm{GL}(k, \mathbb{R}) allow computation of H^1(K; \mathcal{L}) using duality to pair with H_1(K; \mathcal{L}^\vee \otimes \mathbb{Z}_w), revealing non-trivial twists that vanish in the orientable double cover (the torus).^[27]^[19] To address open manifolds, where standard Poincaré duality fails due to lack of compactness, compactly supported versions incorporate Borel-Moore homology: for an open orientable n-manifold M, H^j_c(M; \mathcal{L}) \cong H_{n-j}(M; \mathcal{L}^\vee \otimes \mathrm{or}_M), where H_*^BM denotes homology with infinite chains allowed outside compact sets. This extends naturally to non-orientable cases using \mathbb{Z}_w. In the sheaf-theoretic framework, Verdier duality provides the derived category formulation, asserting that for a smooth manifold M of dimension n, the dualizing complex is \omega_M (the orientation sheaf shifted), yielding RHom(\mathcal{L}, \omega_M ) \cong RHom_c(\mathcal{L}, \underline{k}) in the derived category of sheaves, recovering twisted Poincaré duality upon taking cohomology.^[28]

Geometric and Algebraic Connections

Local systems on smooth manifolds establish a profound connection to differential geometry through flat connections. A local system of rank k corresponds to a representation \rho: \pi_1(M) \to GL(k, \mathbb{R}), where \pi_1(M) is the fundamental group of the manifold M. This representation defines parallel transport, yielding a flat vector bundle whose sections form the local system. By the Riemann-Hilbert correspondence, such flat bundles with integrable connections are equivalent to local systems, enabling the computation of de Rham cohomology with local coefficients as the cohomology of the associated flat bundle.^[29] In algebraic geometry, étale local systems over schemes generalize this framework, corresponding to continuous \ell-adic representations of the étale fundamental group \pi_1^{\text{ét}}(X, \overline{x}) into GL(k, \mathbb{Q}_\ell). These \ell-adic sheaves play a pivotal role in the Langlands program, where they encode geometric analogs of Galois representations and relate to automorphic forms on varieties. For instance, on algebraic curves, étale local systems parametrize the input data for the geometric Langlands correspondence, linking cohomology of moduli stacks to representations of Langlands dual groups. Recent developments in the 2020s, including progress on the categorical geometric Langlands conjecture, have further illuminated these ties through categorical enhancements.^[30] Local systems also intersect with physics in gauge theory, particularly Yang-Mills theories on manifolds, where they model the holonomy of flat connections in the bundle of Lie algebra-valued forms. Wilson lines, defined as path-ordered exponentials \mathcal{P} \exp \left( \int_\gamma A \right) along paths \gamma, compute gauge-invariant observables representing the representation of \pi_1(M) induced by the connection; for pure Yang-Mills, the moduli space of flat connections is precisely the character variety of local systems. This framework extends to topological quantum field theories (TQFTs), where local systems furnish the representation data for constructing extended TQFTs via higher categories, as in the Reshetikhin-Turaev construction for 3-manifolds.^[31]^[32]^[33] A cornerstone linking these geometric and algebraic aspects is Deligne's development of mixed Hodge structures on the cohomology of local systems over complex algebraic varieties. In his seminal work, Deligne equips the hypercohomology \mathbb{H}^*(X, \mathbb{Q} \otimes \mathcal{L}) of a local system \mathcal{L} with a mixed Hodge structure, compatible with the variation over the base and reflecting the transcendental nature of the periods. This structure unifies Betti, de Rham, and Hodge cohomologies for non-constant coefficients, facilitating comparisons between topological and analytic invariants on varieties.

References

[1]
[PDF] Local Systems and Constructible Sheaves - LSU Math
Feb 1, 2007 · A sheaf F on X is locally constant, or F is a local system, if for all x ∈ X, there is a neighborhood U containing x such that F|U is a constant ...
[2]
local system in nLab
### Summary of Local Systems in Mathematics
[3]
Homology With Local Coefficients - jstor
HOMOLOGY WITH LOCAL COEFFICIENTS. By N. E. STEENROD. (Received January 26, 1942). 1. Introduction. In a recent paper [16] the author has had occasion to ...
[4]
Section 6.7 (006S): Sheaves—The Stacks project
The sheaf that assigns to an open U \subset X the set of all locally constant maps U \to A with restriction mappings given by restrictions of functions.<|control11|><|separator|>
[5]
local system in nLab
Apr 10, 2025 · A local system – which is short for local system of coefficients for cohomology – is a system of coefficients for twisted cohomology.
[6]
Section 59.64 (09Y8): Locally constant sheaves—The Stacks project
A locally constant sheaf has a covering where each part is a constant sheaf. A constant sheaf is the sheafification of a presheaf U \mapsto E.
[7]
[PDF] Local Systems and Constructible Sheaves
We study here sheaves of groups with topological interest known as local sys- tems or locally constant sheaves. They arise in mathematics as solutions of linear.
[8]
[PDF] Homology With Local Coefficients
Dec 28, 2012 · Any closed (q + 1)-cell is simply-connected, there is there- fore just one system of local groups defined over it which agrees with a given.
[9]
[PDF] ALGEBRAIC DIFFERENTIAL EQUATIONS - MIT Mathematics
Mar 28, 2025 · In other words, the fiber Lx becomes a representation of π1(X, x); call it the monodromy representation. The image of π1(X, x) → GL(Lx) is ...
[10]
[PDF] arXiv:1507.03144v3 [math.AG] 20 Feb 2016
Feb 20, 2016 · ) → Γ → 1. If V is a local system over X, then one has a monodromy representation π1(X, yo) → Aut Vo where Vo denotes the fiber over yo ...
[11]
[PDF] arXiv:1805.01264v3 [math.AT] 26 Apr 2019
Apr 26, 2019 · If G = π1(X, b) is the fundamental group of a pointed path- connected space (X, b), then representations of G are classical local systems over X ...
[12]
[PDF] arXiv:2101.10705v1 [math.AG] 26 Jan 2021
Jan 26, 2021 · equivalence between locally constant sheaves and representations of the fundamental group ... example if X is path connected and has trivial ...
[13]
[PDF] Lectures on Algebraic Topology - MIT Mathematics
Fiber bundles are naturally occurring objects. For instance, a covering space E → B is precisely a fiber bundle with discrete fibers. Page 141. 42. FIBER ...
[14]
[PDF] Noncommutative Local Systems - arXiv
Nov 10, 2014 · It means that in case of linearly connected space X local systems can be defined by representations of fundamental group π1(X ). Otherwise.
[15]
[PDF] Classical Motivation for the Riemann-Hilbert Correspondance
There is a fundamental link between local systems and vector bundles with connection. The starting point is the following reformulation of the local uniqueness.
[16]
[PDF] Topology and Groupoids
This book is intended as a textbook on point set and algebraic topology at the undergraduate and immediate post-graduate levels. In particular,.
[17]
[1805.02539] Persistent Local Systems - arXiv
May 7, 2018 · In this paper, we give lower bounds for the homology of the fibers of a map to a manifold. Using new sheaf theoretic methods, we show that these lower bounds ...Missing: Sheekman | Show results with:Sheekman
[18]
[PDF] Local Systems and Derived Algebraic Geometry
Mar 10, 2015 · Thus, to give a local system is to give an endomorphism corresponding to γ, i.e. an element of GLpV q. Note that this is equivalent to giving a ...
[19]
[PDF] Sheaves and Geometry - Purdue Math
A function is locally constant if it is constant in a neighbour- hood of a point. The set of locally constant functions, denoted by T(U) or TX(U), is a sheaf.
[20]
[PDF] Algebraic Topology - Cornell Mathematics
This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in ...Missing: monodromy | Show results with:monodromy
[21]
[PDF] arXiv:2001.08243v2 [math.AT] 26 Oct 2020
Oct 26, 2020 · The orientation of a loop provides over RP2 the local integer coefficient system. ̺ : π1(RP2) → Aut(Z) given by ̺(1) = 1 and ̺(−1) = −1 ...
[22]
[PDF] notes on crystals and algebraic d-modules
In other words, the category of vector bundles with flat connection on X is equivalent to the category of local systems of vector spaces on X. Now suppose ...<|control11|><|separator|>
[23]
[PDF] A Primer on Sheaf Theory and Sheaf Cohomology - UT Math
The first few sections introduce some key defi- nitions and technical tools leading to the definition of sheaf cohomology in the language of derived functors.
[24]
[PDF] Computing cohomology of local systems
The proof of both theorems naturally falls into two parts—the first one a local computation of the cohomology of each complex; the second one formal arguments.
[25]
Singular homology and cohomology with local coefficients ... - MSP
In this section we introduce the usual sin- gular homology of X with coefficients in a local system. ... [14] N. E. Steenrod, Homology with local coefficients, ...
[26]
[PDF] Quick and dirty introduction to perverse sheaves - Math (Princeton)
Definition 2.1 We say that a sheaf F ∈ Sh(X, F) is constructible if there exists a stratifica- tion of X such that the restriction of F to each stratum is a ...
[27]
Higher local systems and the categorified monodromy equivalence
We study local systems of (\infty,n)-categories on spaces. We prove that categorical local systems are captured by (higher) monodromy data.
[28]
[PDF] Local coefficients and Poincaré duality - Bena Tshishiku
Jul 15, 2016 · Poincaré duality is an important feature of manifolds that distinguishes them from other topological spaces. The simplest statement (the one ...
[29]
[PDF] VERDIER DUALITY 1. Introduction Let M be a smooth, compact ...
Jul 29, 2011 · The Verdier duality theorem will apply not only to manifolds, but more generally to locally compact spaces of finite cohomological dimension ...
[30]
[PDF] Regular Singular Connections and the Riemann–Hilbert ...
These are defined in terms of flat connections on locally free sheaves. Definition 2. Let M be a connected complex manifold and X a smooth connected complex ...
[31]
[PDF] Lectures on the Langlands Program and Conformal Field Theory
representation π one could always find a Galois representation (or an `-adic local system) ... adic representations of the fundamental group of a curve over a.
[32]
[PDF] Gauge Theory - DAMTP
... Yang-Mills Theory. 26. 2.1 Introducing Yang-Mills. 26. 2.1.1 The Action. 29. 2.1.2 Gauge Symmetry. 31. 2.1.3 Wilson Lines and Wilson Loops. 33. 2.2 The Theta ...
[33]
[PDF] Physics Quantum Field Theory and the Jones Polynomial - People
Such flat connections are completely character- ized by the "Wilson lines", that is, the holonomies around non-contractible loops on Σ. A simple count of ...
[34]
Higher Categories and Topological Quantum Field Theories - arXiv
Oct 24, 2016 · The construction can be extended to obtain a (3+1)-dimensional topological quantum field theory (TQFT).Missing: local systems