Fact-checked by Grok 2 weeks ago

Covering group

In mathematics, a covering group of a topological group H is a topological group G equipped with a continuous surjective homomorphism p: G \to H that serves as a covering map, meaning p is a local homeomorphism with discrete fibers, and the kernel of p is central and prodiscrete (a product of discrete groups).^[1] This structure generalizes the notion of covering spaces from topology to groups, preserving the group operation under the projection. The most prominent example is the universal covering group, which is a simply connected covering group that covers all other connected covering groups of H via further covering homomorphisms, provided H is connected, locally path-connected, and semilocally simply connected. Introduced by Chevalley in the context of Lie groups, the universal covering group \tilde{H} of a connected Lie group H has \pi_1(H) as its kernel and classifies representations of H through those of \tilde{H}, since projective representations of H lift to linear ones on \tilde{H}. For instance, the universal covering group of the special orthogonal group SO(3) is the special unitary group SU(2), a double cover reflecting the fundamental group \mathbb{Z}/2\mathbb{Z}. In the discrete setting of finite group theory, the term covering group often refers to a Schur covering group (or Schur cover), which is a central extension $1 \to M(G) \to \tilde{G} \to G \to 1 of a finite group G by its Schur multiplier M(G) = H^2(G, \mathbb{Z}), the second cohomology group measuring the obstruction to lifting projective representations to ordinary ones.^[2] Every finite group admits a Schur covering group, unique up to isomorphism when G is perfect, and it linearizes all projective representations of G.^[2] For example, the Schur cover of the alternating group A_5 is the binary icosahedral group of order 120, with multiplier \mathbb{Z}/2\mathbb{Z}. These constructions bridge algebraic group theory with representation theory and cohomology.

Definition and Fundamentals

Definition of Covering Groups

A topological group is a mathematical structure consisting of an abstract group G equipped with a topology such that the group multiplication operation G \times G \to G, (g, h) \mapsto gh, and the inversion operation G \to G, g \mapsto g^{-1}, are both continuous maps with respect to the product topology on G \times G.^[3] This topology ensures that the group structure interacts compatibly with the underlying topological space, enabling the study of continuous homomorphisms and other topological properties within group theory.^[3] In the broader context of algebraic topology, a covering space of a topological space B (the base space) is a topological space E (the total space) together with a continuous surjective map p: E \to B known as a covering map. This map is a local homeomorphism, meaning that for every point b \in B, there exists an open neighborhood V of b such that the preimage p^{-1}(V) is a disjoint union of open sets in E, each of which is homeomorphic to V via the restriction of p.^[4] The fibers p^{-1}(b) are discrete, and this structure generalizes the idea of multiple "sheets" layered over the base space in a locally trivial manner.^[4] Building on these concepts, a covering group of a topological group H is defined as a topological group G equipped with a continuous surjective homomorphism \pi: G \to H that serves as a covering map, hence a local homeomorphism, with a discrete kernel. The kernel of \pi, which is the preimage of the identity element in H, must be a discrete normal subgroup of G. When H is connected, this kernel is central, meaning it commutes with every element of G.^[5] This ensures that the group homomorphism preserves the algebraic structure while the covering property guarantees topological compatibility, allowing the fibers to behave discretely and the map to lift paths and homotopies in a controlled way. The notion of covering groups extends the classical theory of covering spaces to the setting of topological groups, formalizing how group extensions can be realized topologically.^[6] It originated in the mid-20th century as part of efforts to generalize covering space constructions to groups with nontrivial topology, with foundational properties explored in early works such as Iwasawa's 1950 study on the structure and uniqueness of such coverings for connected, locally connected topological groups.^[6]

Covering Homomorphisms and Spaces

A covering homomorphism between topological groups G and H is defined as a continuous, open, surjective group homomorphism \pi: G \to H whose kernel is discrete. When H is connected, this kernel is central.^[7] This ensures that G serves as a covering space of H in the topological sense, where the map \pi behaves as a covering map while preserving the group structure.^[8] The openness of \pi follows under mild assumptions, such as G being Lindelöf and locally compact while H is a Baire space, guaranteeing that the discrete kernel implies the map is open.^[7] The covering property manifests through evenly covered neighborhoods: for every h \in H, there exists an open neighborhood U of h such that \pi^{-1}(U) is a disjoint union of open sets in G, each of which is mapped homeomorphically onto U by \pi.^[7] Near the identity, this takes the form \pi^{-1}(W') being homeomorphic to \ker(\pi) \times W', where W' is an open neighborhood of the identity e_H in H, and \ker(\pi) carries the discrete topology.^[7] This local triviality underscores the fiber bundle-like structure of the covering, with the discrete fibers ensuring no pathological branching. As a group homomorphism, \pi is compatible with the multiplicative structures, satisfying \pi(gh) = \pi(g) \pi(h) for all g, h \in G.^[8] This projection of the group operation in G to that in H maintains the algebraic integrity of the covering, allowing lifts of paths and homotopies in H to G while respecting the topology.^[7] The fiber over the identity, \pi^{-1}(e_H), coincides with the kernel of \pi and forms a discrete normal subgroup of G.^[8] When H is connected, since the kernel is central, it commutes with every element of G, a property exemplified in classical cases like the double cover \rho: \operatorname{[Spin](/page/Spin)}(k) \to \operatorname{SO}(k, \mathbb{R}), where the kernel is \{\pm I\}.^[7] This central discreteness ensures the covering is "universal" in simply connected contexts, though here it strictly delineates the homomorphism's role in bridging the spaces.^[8]

Core Properties and Structures

Algebraic Properties

A covering group G of a group H arises as a central extension $1 \to K \to G \xrightarrow{\pi} H \to 1, where K = \ker(\pi) is a discrete abelian subgroup contained in the center Z(G) of G. This extension is central, meaning that K commutes elementwise with all elements of G, and it is universal in the algebraic sense that its projection to H factors uniquely through any other central extension of H by a quotient of K.^[9] For finite groups H, such covering groups exist and are finite, providing a stem extension where K is also contained in the derived subgroup G' of G. In the case of discrete groups, the kernel K is identified with the Schur multiplier M(H) of H, defined as the second integral homology group H_2(H, \mathbb{Z}). Central extensions of H by a fixed abelian group A are classified by the second cohomology group H^2(H, A).^[10] The Schur multiplier M(H) = H_2(H, \mathbb{Z}) is the kernel of the universal central extension of H, corresponding to a canonical class in H^2(H, M(H)). The covering group G serves as this universal central extension (or stem cover) in the discrete case. The relation stems from Schur's original work on projective representations, where M(H) measures the obstructions to lifting linear representations of H to those of extensions. Seminal results establish that for perfect groups (where H = H'), the covering group is unique up to isomorphism, while for general finite groups, there may be multiple non-isomorphic covering groups, though a canonical stem cover exists. The short exact sequence $1 \to M(H) \to G \to H \to 1 encapsulates these algebraic properties, with M(H) acting as the fundamental obstruction in extension theory. This sequence is nonsplit in general, reflecting the nontriviality of the extension class in the second cohomology group H^2(H, M(H)). For example, the double cover of the alternating group A_5, which is the binary icosahedral group $2 \cdot A_5 \cong SL(2,5), illustrates how M(A_5) \cong \mathbb{Z}/2\mathbb{Z} yields a central extension that is perfect and stem. Regarding multiplicativity, the Schur multiplier of a direct product H = G_1 \times G_2 of finite groups satisfies M(H) \cong M(G_1) \times M(G_2) \times (G_1^{\mathrm{ab}} \wedge G_2^{\mathrm{ab}}), where G_i^{\mathrm{ab}} = G_i / G_i' is the abelianization of G_i and \wedge denotes the exterior product of abelian groups. Consequently, the covering group of the direct product can be constructed as a central extension by this larger multiplier, though it does not always decompose as a direct product of the individual covering groups unless one of the factors is abelian (in which case M(H) \cong M(G_1) \times M(G_2)). This structure highlights how covering groups interact with direct products under conditions where the abelianizations are trivial, such as for perfect groups.

Topological and Group Compatibility

In the context of covering groups, the topological structure of the total space G interacts seamlessly with its algebraic group operations, ensuring that the covering homomorphism \pi: G \to H is both a continuous open map and a group homomorphism. The multiplication map m: G \times G \to G and the inversion map i: G \to G are continuous with respect to the topologies on G and H, as \pi lifts the group operations from H while preserving their continuity. Specifically, the group structure on G is defined such that for any g_1, g_2 \in G, the product g_1 \cdot g_2 is the unique endpoint of the path lift in G of the constant path at \pi(g_1) followed by the path from \pi(g_1) to \pi(g_1) \cdot \pi(g_2) in H, starting at g_1; this construction guarantees continuity because the path-lifting property of covering maps ensures unique and continuous lifts. Similarly, inversion is defined via lifting the path from the identity to \pi(g)^{-1}, maintaining topological compatibility. This compatibility holds because \pi is an open epimorphism with discrete kernel, making G a topological group whose operations align with those of H. The covering \pi: G \to H is locally trivial in the sense of fiber bundles, with the fiber K = \ker(\pi) being a discrete normal subgroup of G. For every point h \in H, there exists an open neighborhood U \subset H such that \pi^{-1}(U) is homeomorphic to the disjoint union of copies of U \times K, where K carries the discrete topology. This local product structure positions the covering as a principal K-bundle over H, where K acts freely and continuously on G by left multiplication: for k \in K and g \in G, the map g \mapsto k \cdot g is a homeomorphism of G fixing fibers setwise, and the action is free since k \cdot g = g implies k = e (the identity). The normality of K ensures the quotient G/K \cong H inherits the group structure compatibly, while the discreteness of K aligns the bundle topology with the covering space properties. In cases where K is central (as is typical for the algebraic extensions underlying covering groups), this action further commutes with the right translations, enhancing the compatibility between the bundle and group structures.^[11] A key topological feature preserved in covering groups is the path-lifting property, analogous to that of general covering spaces but enriched by the group structure. Any path \gamma: [0,1] \to H with \gamma(0) = h lifts uniquely to a path \tilde{\gamma}: [0,1] \to G starting at any chosen \tilde{h} \in \pi^{-1}(h), and the endpoint \tilde{\gamma}(1) determines the homotopy class of \gamma relative to the basepoint. For loops based at the identity e \in H, the lifted path starting at e \in G ends at an element of K, inducing a surjection from the set of homotopy classes of such loops onto the elements of K (via monodromy), with kernel p_* \pi_1(G, e), establishing the isomorphism \pi_1(H, e) / p_* \pi_1(G, e) \cong K.^[1] This reflects the central role of the kernel in encoding (a quotient of) the fundamental group information. This setup is facilitated by the fact that the fundamental group of any topological group is abelian, ensuring the kernel K is central.^[1] Regarding compactness, the covering structure preserves it under finite-sheeted conditions: if H is compact and K is finite (equivalently, the covering is finite-sheeted, as the number of sheets equals |K|), then G is compact. This follows because G decomposes as a finite disjoint union of open sets each homeomorphic via \pi to open covers of the compact space H, and since \pi is a local homeomorphism and open, the preimage of a compact set under a finite-sheeted covering is compact. The finite index of K in G (corresponding to the order of H, but constrained by compactness) ensures this preservation, contrasting with infinite-sheeted cases where G may fail to be compact despite H's compactness.

Constructions and Universal Objects

Inducing Group Structure on Covering Spaces

Given a covering space p: E \to H of a path-connected, locally path-connected topological group H, there exists a unique topological group structure on E (with a chosen basepoint e \in p^{-1}(1_H) as the identity) such that p becomes a continuous group homomorphism. This structure makes (E, p) a covering group of H, where the kernel of p is precisely the discrete fiber p^{-1}(1_H). The existence requires that the deck transformation group \Deck(p) of the covering acts compatibly on E, preserving the lifted group operations derived from H; this compatibility holds automatically under the topological assumptions on H, as the covering map interacts well with the continuous multiplication and inversion in H.^[12]^[13] The construction leverages the unique lifting property of covering maps. To define the group operations on E, first fix the identity element as e. The inversion map i_E: E \to E is obtained as the unique continuous lift of the inversion i_H: H \to H, h \mapsto h^{-1}, such that p \circ i_E = i_H \circ p; this lift exists because p is a covering and i_H is continuous, with the lift specified uniquely over a neighborhood of e by the local triviality of the covering. Similarly, the multiplication m_E: E \times E \to E is the unique continuous lift of m_H: H \times H \to H, the multiplication in H, via the product covering p \times p: E \times E \to H \times H, ensuring p \circ m_E = m_H \circ (p \times p); again, uniqueness follows from specifying the lift over the identity fiber and extending by connectedness of E. These operations are compatible with the topology on E, yielding a topological group.^[12] An algorithmic approach to defining the product explicitly uses path lifting, assuming H admits path spaces. Select a path \delta: [0,1] \to E from e to y \in E. Project this to \beta = p \circ \delta from $1_H to p(y) in H. Left-translate \beta by p(x) to form the path \mu(t) = p(x) \cdot \beta(t) from p(x) to p(x) \cdot p(y) in H. Lift \mu uniquely to a path in E starting at x (using the covering path lifting property), and define x \cdot y as the endpoint of this lift. Associativity and other group axioms follow from the homotopy invariance of path concatenation and the uniqueness of lifts relative to endpoints. This path-based method aligns with the global lifting construction and ensures continuity.^[13] The induced group structure on E is unique up to isomorphism of topological groups over H (i.e., commuting with p) once the identity is fixed. If the covering is regular—meaning the subgroup p_* \pi_1(E, e) \leq \pi_1(H, 1_H) is normal, so \Deck(p) acts transitively and normally—the isomorphism class is absolute, without dependence on the choice of basepoint in the fiber. In general cases, different choices of identity in the fiber yield isomorphic structures via deck transformations.^[12]

Universal Covering Groups

The universal covering group \tilde{H} of a connected topological group H is defined as a simply connected topological group equipped with a continuous covering homomorphism \pi: \tilde{H} \to H that is a local homeomorphism and preserves the group operation, such that the kernel \ker(\pi) is a discrete central subgroup isomorphic to the fundamental group \pi_1(H, e).^[14]^[15] This kernel arises from the action of loops based at the identity element e \in H on the covering space, ensuring that \tilde{H} captures the simply connected essence of H while projecting onto it via the covering map. The simply connectedness of \tilde{H} means its fundamental group is trivial, making it the "maximal" simply connected extension in the category of topological groups. Existence of the universal covering group is guaranteed for connected, locally path-connected topological groups H that are semilocally simply connected, a condition ensuring that sufficiently small neighborhoods around the identity have contractible loops.^[14] The construction proceeds by first forming the universal covering space \tilde{X} of the underlying topological space X of H, which exists under these hypotheses as the space of homotopy classes of paths starting at a basepoint, with the covering map sending each class to its endpoint.^[14] The group structure on \tilde{H} is then induced by lifting the multiplication in H via path concatenation: for lifts \tilde{g}, \tilde{h} \in \tilde{H} of g, h \in H, their product \tilde{g} \cdot \tilde{h} is the lift of gh starting at the appropriate basepoint, ensuring compatibility with the covering homomorphism.^[15] This yields a topological group whose covering map \pi has \ker(\pi) \cong \pi_1(H, e) as discrete central subgroups. Uniqueness holds up to isomorphism of topological groups: any other simply connected covering group of H is isomorphic to \tilde{H}, and the covering homomorphism factors uniquely through \pi via a commutative diagram of group homomorphisms.^[14]^[15] This universal property follows from the lifting criterion for covering spaces and the correspondence between subgroups of \pi_1(H, e) and connected covering spaces, where the trivial subgroup corresponds to the simply connected case. For instance, in the case of the special orthogonal group SO(3), the universal covering group is the double cover SU(2), with kernel \mathbb{Z}/2\mathbb{Z} \cong \pi_1(SO(3)).^[14]

Advanced Structures and Lattices

Lattice of Covering Groups

The set of all covering groups of a given topological group H forms a partially ordered set (poset), where one covering group G' precedes another G (denoted G' \leq G) if there exists a continuous covering homomorphism G \to G' that commutes with the projections to H. This ordering reflects the refinement of covers: larger elements in the poset correspond to finer (higher-degree) coverings of H. This poset is in fact a complete lattice, with the universal covering group of H serving as the maximum element and the identity map on H (the trivial covering group) as the minimum element. The meet of any two covering groups G_1 \to H and G_2 \to H is given by their fiber product G_1 \times_H G_2 \to H, which inherits a topological group structure as the pullback in the category of topological groups. Joins exist for arbitrary subsets via categorical constructions preserving the covering and group properties, ensuring every chain and subfamily has suprema and infima. Up to isomorphism, the covering groups of H are classified by the closed subgroups of the kernel of the universal covering map, which is isomorphic to the fundamental group \pi_1(H, e). Specifically, for each closed subgroup K \leq \pi_1(H, e), the corresponding covering group is the quotient of the universal cover by K, yielding a bijection between such subgroups and isomorphism classes of covering groups. The lattice is complete, as every ascending or descending chain of covering groups admits suprema and infima constructed via the universal properties of pullbacks and the subgroup lattice structure. However, it is not always modular; counterexamples arise in non-abelian cases, such as when the subgroup lattice of \pi_1(H, e) fails modularity.

Covering Groups for Lie Groups

In the context of Lie groups, a covering group of a connected Lie group H is a Lie group G equipped with a surjective Lie group homomorphism \pi: G \to H that serves as a covering map, meaning \pi is a local diffeomorphism. The kernel of \pi is a discrete central subgroup of G, ensuring compatibility with the smooth structure of the groups. This setup preserves the Lie algebra isomorphism between G and H, as the differential of \pi at the identity is an isomorphism of Lie algebras.^[16] The universal covering group of a connected Lie group H is the unique (up to isomorphism) simply connected Lie group \tilde{H} having the same Lie algebra as H, with the covering map \pi: \tilde{H} \to H being the universal cover in the topological sense. For instance, the universal covering group of the circle group S^1 is the additive group \mathbb{R}, and the universal covering group of the projective special linear group \mathrm{PSL}(2,\mathbb{R}) is the infinite cover of the special linear group \mathrm{SL}(2,\mathbb{R}), reflecting the fundamental group \mathbb{Z} of \mathrm{SL}(2,\mathbb{R}). This construction ensures that \tilde{H} captures the full simply connected component while projecting onto H.^[16] For semisimple Lie groups, covering groups connect to the simply connected form through the adjoint representation, where the simply connected cover corresponds to the Lie group whose Lie algebra has the given root system, classified by Dynkin diagrams. The adjoint form arises as a quotient by the finite center, and intermediate covers are determined by subgroups of the center, linking the topology to the algebraic structure encoded in the root system and its Dynkin diagram.^[17] In the case of compact connected Lie groups, the universal covering group is itself compact, and the covering map has finite degree, with the kernel being a finite central subgroup. A prominent example is the double cover \mathrm{Spin}(n) \to \mathrm{SO}(n) for n \geq 3, where \mathrm{Spin}(n) is the simply connected cover realizing the full spin representation. The explicit classification of these covers relies on the Weyl group acting on the root system, determining the possible central extensions and the structure of the fundamental group.^[18]^[19]

Examples and Applications

Classical Examples

One of the most fundamental examples of a covering group arises in the context of the circle group U(1), which is topologically equivalent to the unit circle S^1. The universal covering group of U(1) is the additive group of real numbers \mathbb{R}, with the covering homomorphism given by the exponential map \exp: \mathbb{R} \to U(1), x \mapsto e^{2\pi i x}.^[20] This map is a surjective group homomorphism with kernel \mathbb{Z}, which is infinite cyclic, making \mathbb{R} the simply connected cover that captures all projective representations of U(1).^[20] In the realm of orthogonal groups, the spin groups provide classic instances of covering groups. Specifically, the group \mathrm{Spin}(3) is isomorphic to \mathrm{SU}(2) and serves as a double cover of \mathrm{SO}(3), the special orthogonal group in three dimensions, via the adjoint representation that identifies rotations with conjugations by unit quaternions.^[21] More generally, for n \geq 3, the spin group \mathrm{Spin}(n) is the universal covering group of \mathrm{SO}(n), with the covering map being a double cover having kernel \mathbb{Z}/2\mathbb{Z}; this structure ensures \mathrm{Spin}(n) is simply connected while faithfully representing the Lie algebra of \mathrm{SO}(n).^[22] For symmetric and alternating groups, Schur covering groups exemplify finite covering constructions. For n \geq 4, n \neq 6, 7, the Schur double cover $2 \cdot A_n of the alternating group A_n is a central extension with kernel \mathbb{Z}/2\mathbb{Z}, arising from the Schur multiplier of A_n, and it classifies the projective representations of A_n.^[23] This cover is unique up to isomorphism and stems from the universal central extension properties of perfect groups.^[24] In the discrete finite group setting, covering groups often coincide with stem covers, which are central extensions where the kernel intersects the derived subgroup non-trivially. A prominent example is \mathrm{SL}(2,5), the special linear group of $2 \times 2 matrices over the field with five elements, which is the Schur cover (double cover) of the alternating group A_5 with kernel \mathbb{Z}/2\mathbb{Z}; this extension has order 120 and is the unique non-split central extension realizing the Schur multiplier of A_5.^[25] Unlike connected Lie groups, disconnected topological groups generally lack a universal covering group that is itself a topological group without imposing additional structure, such as restricting to path components or assuming local connectedness; instead, covers may be constructed componentwise, but the full group structure complicates the existence of a simply connected total space.

Applications in Group Theory and Topology

In representation theory, projective representations of a finite group H correspond to ordinary linear representations of its covering group G, allowing the linearization of projective actions and resolving limitations in Schur's lemma for irreducibility over fields like the complex numbers. This lifting is facilitated by the Schur multiplier, which measures the obstruction to such extensions, and ensures that every irreducible projective representation of H arises from an irreducible representation of a minimal covering group, such as the Schur covering group. For instance, Schur's foundational work established that projective representations of symmetric groups S_n lift to linear representations of their double covers for n \geq 4, enabling a complete character theory for these structures.^[24] Covering groups play a key role in homotopy theory and cohomology by encoding the fundamental group \pi_1(H) of a topological group H as the kernel of the covering map to its universal cover, which is simply connected and computes higher homotopy invariants. In group cohomology, central extensions corresponding to covering groups are classified by the second cohomology group H^2(H, \mathbb{Z}), providing a topological measure of extension classes that relate to the structure of classifying spaces BH. This connection is essential in algebraic topology, where covering groups facilitate the computation of cohomology rings for spaces with non-trivial fundamental groups, such as in the study of fibrations and Postnikov towers.^[26]^[27] In quantum mechanics, the spin groups, such as Spin(3) \cong SU(2), serve as double covers of the rotation group SO(3), enabling representations with half-integer spins like s = 1/2 for fermions such as electrons, which are impossible in SO(3) due to its integer spin restriction. These half-integer representations, of dimension $2s + 1, describe the intrinsic angular momentum of particles and arise from the two-to-one homomorphism SU(2) \to SO(3), where a 360-degree rotation corresponds to a phase factor in the spinor space. In gauge theories, universal covering groups ensure the correct topological structure for principal bundles, with the fundamental group \pi_1(G) influencing quantization conditions like the Chern-Simons level k, which must be adjusted by factors related to the covering (e.g., multiples of 4 for SO(3) versus SU(2)).^[28]^[29] Modern applications extend to string theory, where loop groups LG of compact Lie groups G act as central extensions or covers, with their Kac-Moody extensions providing the string class obstruction for lifting bundles over loop spaces, crucial for anomaly cancellation and the geometry of string backgrounds. The string group, modeled via 2-groups from loop group extensions, resolves topological issues in higher-dimensional supergravity and M-theory compactifications. In computational group theory, software like GAP implements the cohomolo package to construct covering groups via Schur multipliers, enabling explicit computations of extensions for finite groups up to moderate orders and supporting algorithmic classification in research on permutation groups and their representations.^[30]^[31]

References

[1]
https://arxiv.org/pdf/1808.08514.pdf
[2]
[PDF] metaplectic analogues of gelfand–graev models
The Schur multiplier of a group G is the group M(G) = H2(G, Z) ≃ H2(G, C×). Definition 2.6. Let G be a finite group. A Schur covering group (or universal ...
[3]
Topological Groups - an overview | ScienceDirect Topics
A topological group is defined as a group equipped with a topology such that the group operations of multiplication and inversion are continuous.
[4]
[PDF] Covering Spaces - UChicago Math
Roughly speaking, a space Y is called a covering space of X if Y maps onto X in a locally homeomorphic way, so that the pre-image of every point in X has the ...
[5]
[1808.08514] On Generalized Covering Groups of Topological Groups
Aug 26, 2018 · It is well-known that a homomorphism p between topological groups K, G is a covering homomorphism if and only if p is an open epimorphism with ...
[6]
On some properties of covering groups of a topological group.
June, 1950 On some properties of covering groups of a topological group. There is no online version at this time. Please download the PDF instead.
[7]
[PDF] When is a group homomorphism a covering homomorphism?
Aug 22, 2007 · If ϕ is a continuous homomorphism from a topological group G to a topo- logical group H, in order that ϕ is a covering homomorphism it is ...
[8]
[PDF] On Generalized Covering Groups of Topological Groups - arXiv
Aug 26, 2018 · Definition 2.1. Let G be a topological group. By a generalized covering group of G, a pair ( eG, p) is composed of a topological group ...
[9]
[PDF] Chapter 3 - Vector bundles and principal bundles
Over a compact Hausdorff space, any vector bundle embeds in a trivial bundle. ... But we'd like to provide a construction to cover arbitrary topological groups.<|control11|><|separator|>
[10]
Existence and uniqueness of group structures on covering spaces ...
Aug 6, 2025 · The covering space endowed with that group structure is a topological group, and a given covering mapping turns into a homomorphism of compact ...
[11]
[PDF] Algebraic topology (Math 592): Problem set 6
Let G be a topological group, and let p : H → G be a covering space. Using the lifting property, show that H admits a unique topological group structure making ...
[12]
None
Below is a merged summary of the universal covering spaces from Hatcher’s *Algebraic Topology* (Chapter on Covering Spaces), consolidating all the information provided across the segments into a single, comprehensive response. To maximize density and clarity, I will use a table format where appropriate to organize key details, followed by a narrative summary that integrates additional insights and nuances. The response retains all information mentioned, including definitions, existence, uniqueness, groups, and useful URLs, while avoiding redundancy where possible.
[13]
https://www.math.ias.edu/~bhatt/teaching/mat592w15/hw6.pdf
[14]
[PDF] Introduction to Lie Groups and Lie Algebras Alexander Kirillov, Jr.
... Lie group then its universal cover ˜G has a canonical structure of a Lie group such that the covering map p: ˜G → G is a morphism of Lie groups, and Ker p = π1( ...
[15]
Modular lattice - Wikipedia
The lattice of normal subgroups of a group is modular. But in general the lattice of all subgroups of a group is not modular. For an example, the lattice of ...
[16]
[PDF] Lie groups and Lie algebras (Winter 2024)
If Γ ⊆ π1(G) is any subgroup, then Γ (viewed as a subgroup of eG) is central, and so eG/Γ is a Lie group covering G, with π1(G)/Γ as its group of deck ...
[17]
[PDF] the classification of simple complex lie algebras - UChicago Math
Aug 24, 2012 · ... groups to classifying simple simply-connected Lie groups ... Dynkin diagram corresponds to the root system of a unique semisimple Lie algebra.
[18]
[PDF] Lectures on Lie Groups Dragan Milicic - math.utah.edu
D is compact and C discrete, D ∩ C is finite. D. Let G be a connected compact Lie group and ˜G its universal covering group. Let p : ˜G −→ G be the ...<|control11|><|separator|>
[19]
[PDF] Examples of Lie Groups in Geometry and Topology
Dec 8, 2022 · The universal cover ˜G of G is a Lie group in a natural way. It is isomorphic (as a manifold) to a product H × Rn where H is a compact ...
[20]
[PDF] The Universal Covering Group of U(n) and Projective Representations
Using fibre bundle theory we construct the universal covering group of U(n), ˜U(n), and show that ˜U(n) is isomorphic to the semidirect product SU(n) s R.
[21]
[PDF] SU(2)'s Double-Covering of SO(3)
Dec 12, 2019 · We say that SU(2) double-covers SO(3). Since the map U → R is 2-to-1, not 1-to-1, SO(3) and SU(2) clearly can't be isomorphic groups.
[22]
[PDF] PQM Supplementary Notes: Spin, topology, SU(2)→SO(3) etc
the rotation group—in our new terminology, we have a projective representation of SO(3). 3 The Double Cover SU(2) → SO(3). It is not too difficult to live ...
[23]
Essential dimension of double covers of symmetric and alternating ...
Jun 9, 2019 · Title:Essential dimension of double covers of symmetric and alternating groups ; I. Schur studied double covers \widetilde{\Sym}^{\pm}_n$\ ...
[24]
[PDF] Representations of the covering groups of the symmetric groups and ...
Theorem 1.2 (Schur [S3]) For n ≥ 4, there are two representation groups of. Sn, which are isomorphic only for n = 6. Both representation groups can be given ...
[25]
ATLAS: Alternating group A5 - Atlas of Finite Group Representations
Apr 23, 1999 · Standard generators of the double cover 2.A5 (or SL2(5)) are preimages A and B where B has order 3 and AB has order 5.
[26]
[PDF] classification of group extensions and h2 - UChicago Math
Aug 26, 2011 · Group Extensions and Group Cohomology. 3.1. Group extensions. Definition 3.1. Let A an abelian group and G be any group. We say a third group.
[27]
[PDF] covering groups of non-connected topological groups - Ronald Brown
We deduce that any topological group X admits a simply connected covering group covering all the components of X (Corollary 5.6). According to comments in [20], ...
[28]
[PDF] Quantum Theory, Groups and Representations: An Introduction ...
The intent was to cover the basics of quantum mechanics, up to and including relativistic quantum field theory of free fields, from a point of view ...
[29]
Physics Topological Gauge Theories and Group Cohomology
These topological "spin" theories will require a definite choice of spin structure on the manifold in order to be well-defined.
[30]
Loop Groups, Higgs Fields and Generalised String Classes - arXiv
Jun 26, 2009 · We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose ...Missing: covering | Show results with:covering
[31]
cohomolo - GAP packages
Feb 20, 2024 · The cohomolo package is a GAP interface to some C programs for computing Schur multipliers and covering groups of finite groups and first and second cohomology ...