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Parallelizable manifold
Parallelizable manifold
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In mathematics, a differentiable manifold of dimension n is called parallelizable[1] if there exist smooth vector fields on the manifold, such that at every point of the tangent vectors provide a basis of the tangent space at . Equivalently, the tangent bundle is a trivial bundle,[2] so that the associated principal bundle of linear frames has a global section on

A particular choice of such a basis of vector fields on is called a parallelization (or an absolute parallelism) of .

Examples

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  • An example with is the circle: we can take V1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangent directions at each point. More generally, every Lie group G is parallelizable, since a basis for the tangent space at the identity element can be moved around by the action of the translation group of G on G (every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in G).
  • A classical problem was to determine which of the spheres Sn are parallelizable. The zero-dimensional case S0 is trivially parallelizable. The case S1 is the circle, which is parallelizable as has already been explained. The hairy ball theorem shows that S2 is not parallelizable. However S3 is parallelizable, since it is the Lie group SU(2). The only other parallelizable sphere is S7; this was proved in 1958, by Friedrich Hirzebruch, Michel Kervaire, and by Raoul Bott and John Milnor, in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the normed division algebras of the real numbers, complex numbers, quaternions, and octonions, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires algebraic topology.
  • The product of parallelizable manifolds is parallelizable.
  • Every orientable closed three-dimensional manifold is parallelizable.[3]

Remarks

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  • Any parallelizable manifold is orientable.
  • The term framed manifold (occasionally rigged manifold) is most usually applied to an embedded manifold with a given trivialisation of the normal bundle, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the tangent bundle.
  • A related notion is the concept of a π-manifold.[4] A smooth manifold is called a π-manifold if, when embedded in a high dimensional euclidean space, its normal bundle is trivial. In particular, every parallelizable manifold is a π-manifold.

See also

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Notes

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References

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Parallelizable manifold

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In , a parallelizable manifold is a smooth differentiable manifold of nn whose is trivial, meaning it is isomorphic to the product bundle M×RnM \times \mathbb{R}^n, or equivalently, it admits nn global smooth vector fields that are linearly independent at every point. This property implies the existence of a global framing of the , allowing for a consistent choice of basis vectors across the entire manifold without singularities. Prominent examples of parallelizable manifolds include Euclidean space Rn\mathbb{R}^n, which is trivially parallelizable via its standard coordinate vector fields. The nn-dimensional torus TnT^n is parallelizable, as it inherits a framing from the product structure of circles, each of which is parallelizable. All Lie groups, such as the general linear group GL(n,R)GL(n, \mathbb{R}), special unitary group SU(n)SU(n), and orthogonal group O(n)O(n), are parallelizable due to their left-invariant vector fields providing a global frame. Among spheres, S1S^1, S3S^3, and S7S^7 are parallelizable—the former via its structure as unit complex numbers, the latter two leveraging their structures as unit quaternions and octonions, respectively—while higher odd-dimensional spheres like S5S^5 and even-dimensional ones like S2kS^{2k} are not. Additionally, by Stiefel's theorem, every compact orientable 3-manifold is parallelizable. Parallelizable manifolds exhibit several notable properties, including , since volume forms are differential nn-forms that are sections of the bundle ΛnT∗M\Lambda^n T^*M, the nnth exterior power of the cotangent bundle T∗MT^*M; the triviality of the TMTM implies the triviality of T∗MT^*M (as the dual of a trivial bundle is trivial; this is because a trivial bundle, represented as π:M×V→M\pi: M \times V \to M, has a dual bundle that can be represented as π:M×V∗→M\pi: M \times V^* \to M), and thus ΛnT∗M\Lambda^n T^*M is also trivial, allowing for a global nowhere-vanishing . In low dimensions, all orientable 1-dimensional manifolds are parallelizable, and among compact closed orientable 2-dimensional manifolds, only the is. Proof: If SS is a parallelizable surface, then it admits a flat Riemannian metric—that is, one where it is possible to find a frame in which all the components of the metric tensor g are constant—just choose a smooth global frame and declare it to be orthonormal. If in addition SS is compact, the implies that it has Euler characteristic zero. The only compact orientable surface with χ(S)=0\chi(S)=0 is the torus.; in dimension 3, the property holds precisely for orientable compact ones. Beyond dimension 7, no spheres are parallelizable, and in general, parallelizability imposes strong topological constraints, such as vanishing Stiefel–Whitney classes. These manifolds play a key role in and , facilitating studies of framings, exotic structures, and algebraic extensions like nearly parallel G2G_2-structures.

Definition and Fundamentals

Definition

A smooth manifold MM of dimension nn comes equipped with a TMTM, which is a smooth π:TM→M\pi: TM \to M whose fiber over each point p∈Mp \in M is the TpMT_p M, a real isomorphic to Rn\mathbb{R}^n. Such a manifold MM is parallelizable if its tangent bundle TMTM is trivial, meaning there exists a smooth bundle isomorphism TM≅M×RnTM \cong M \times \mathbb{R}^n. Note that while the triviality of the tangent bundle implies that its total space is diffeomorphic to M×RnM \times \mathbb{R}^n, the definition of parallelizability requires a bundle isomorphism preserving the vector bundle structure, not merely a diffeomorphism of the total spaces as manifolds. This triviality is equivalent to the existence of nn smooth vector fields X1,…,XnX_1, \dots, X_n on MM that are linearly independent at every point, thereby providing a global frame for the tangent spaces. The term "parallelizable" derives from the ability to "parallelize" the tangent spaces globally without twisting, enabling a consistent identification of bases across the manifold.

Tangent Bundle Triviality

A smooth manifold MM of nn is parallelizable its TMTM is trivial, meaning there exists a TM≅εnTM \cong \varepsilon^n, where εn=M×Rn\varepsilon^n = M \times \mathbb{R}^n denotes the trivial bundle of rank nn over MM. This equivalence holds because the triviality of TMTM is characterized by the existence of nn global sections that form a basis for each TpMT_pM. The triviality of the has significant structural consequences. It guarantees the existence of nn global nowhere-vanishing vector fields on MM that span TpMT_pM at every point p∈Mp \in M, providing a global frame for the spaces. Moreover, this allows for the definition of a global in the sense of a consistent moving frame, enabling the expression of vectors uniformly across MM without local obstructions. To construct the trivialization explicitly, suppose {X1,…,Xn}\{X_1, \dots, X_n\} is a global frame of nowhere-vanishing vector fields on MM that are linearly independent at each point. Define a bundle map ϕ:TM→εn\phi: TM \to \varepsilon^n by ϕp(v)=(p,(a1,…,an))\phi_p(v) = (p, (a_1, \dots, a_n)) for v∈TpMv \in T_pM, where v=∑i=1naiXi(p)v = \sum_{i=1}^n a_i X_i(p). This map is a fiberwise linear isomorphism, as the frame spans each tangent space, yielding the desired bundle isomorphism TM≅εnTM \cong \varepsilon^n. Parallelizability depends on the existence of such for the manifold's nn, and thus holds independently of the specific value of nn whenever the frame can be constructed; the triviality is a property intrinsic to the bundle structure for that dimension.

Characterizations

Topological Characterizations

A fundamental topological obstruction to parallelizability arises from the . For a compact manifold MM, the existence of a global framing of the TMTM implies that the e(TM)∈Hn(M;Z)e(TM) \in H^n(M; \mathbb{Z}) vanishes, since the Euler class of a trivial bundle is zero. Consequently, the χ(M)=⟨e(TM),[M]⟩=0\chi(M) = \langle e(TM), [M] \rangle = 0, where [M][M] denotes the fundamental class of MM. This condition is necessary but not sufficient in dimensions greater than 1. More comprehensive topological characterizations involve the vanishing of characteristic classes associated to the . Specifically, all Stiefel-Whitney classes wi(TM)∈Hi(M;Z/2Z)w_i(TM) \in H^i(M; \mathbb{Z}/2\mathbb{Z}) must vanish for i≥1i \geq 1, as these classes are invariants of the bundle's equivalence class, and the total Stiefel-Whitney class of a trivial bundle is 1. The first Stiefel-Whitney class w1(TM)=0w_1(TM) = 0 ensures , while higher classes wi(TM)=0w_i(TM) = 0 for i≥2i \geq 2 provide further obstructions detectable in . For orientable manifolds, the pk(TM)∈H4k(M;Z)p_k(TM) \in H^{4k}(M; \mathbb{Z}) also vanish, since they are defined via the of TMTM and the Chern classes of a trivial complex bundle are trivial, yielding pk=(−1)kc2k(TM⊗C)p_k = (-1)^k c_{2k}(TM \otimes \mathbb{C}). These vanishing conditions are necessary for TMTM to be trivial as a topological . Parallelizability equates to the triviality of TMTM over the topological category. This holds because topological vector bundles over manifolds can be trivialized precisely when their classifying map to the is nullhomotopic in the topological sense. Parallelizability relates closely to stable triviality of the , where TM⊕ϵkTM \oplus \epsilon^k is trivial for some k≥0k \geq 0, but full parallelizability requires k=0k=0. In the smooth category, the space of smooth framings on a given may differ from the topological framings due to exotic smooth structures; for instance, the framed groups Rn\mathcal{R}_n classify the difference, with non-trivial elements corresponding to spheres that obstruct smooth parallelizations even when topological ones exist. Kervaire and Milnor computed these groups, showing that in dimensions like 10, certain topological manifolds admit topological framings but no smooth ones.

Analytic Characterizations

Analytic characterizations of parallelizable manifolds often rely on differential forms and operators to detect the triviality of the through cohomological and integrability conditions. In the context of , the vanishing of certain cup products in H∗(M,R)H^*(M, \mathbb{R}) corresponds to the absence of characteristic classes, confirming triviality via the de Rham theorem equating smooth cohomology to singular with real coefficients. For instance, in dimension 3, ensures parallelizability. The Frobenius theorem connects to parallelizability through the integrability of distributions on the manifold. A trivial admits a flat connection supporting global parallel sections, and the Frobenius theorem ensures local integrability of the corresponding system, which extends globally on parallelizable manifolds where the distribution is the full TMTM and thus involutive by definition. This local-to-global extension via Frobenius provides an analytic criterion: the existence of a nowhere-vanishing frame requires the bracket of local sections to remain within the distribution, a condition automatically satisfied for trivial bundles but used to construct parallel sections algebraically on manifolds like tori or groups. In even dimensions, parallelizable manifolds admit almost complex structures, as the trivial tangent bundle of rank $2nsupportsacompatible[endomorphism](/page/Endomorphism)supports a compatible [endomorphism](/page/Endomorphism)JwithwithJ^2 = -\mathrm{id}.TherelationtoparallelizabilityisanalyticviatheNijenhuistensor. The relation to parallelizability is analytic via the Nijenhuis tensor N_J,whichmeasuresnon−integrability;forastandardform, which measures non-integrability; for a standard form Jonaparallelizablemanifoldon a parallelizable manifoldM(e.g.,(e.g.,J v_i = w_i,, J w_i = -v_iforaglobalframefor a global frame{v_i, w_i}),theJ−invariantstructureensures), the J-invariant structure ensures N_J = 0ifandonlyifif and only ifJisintegrable,butevennon−integrablecasesyieldpseudoholomorphictrivialityofthecanonicalbundlewhenis integrable, but even non-integrable cases yield pseudoholomorphic triviality of the canonical bundle when\bar{\partial} \psi = 0foracanonicalsectionfor a canonical section\psi,linkingtoKodairadimension, linking to Kodaira dimension \mathrm{kod}(M) = 0.Thisprovidesanobstruction:non−vanishing. This provides an obstruction: non-vanishing \bar{\partial}$-operator terms detect deviations from complex parallelizability in even-dimensional settings. Index theory offers analytic hints for non-parallelizability through the Atiyah-Singer index theorem applied to Dirac operators on spin manifolds. The index, computed via local densities and -genus, can yield nonzero values indicating non-triviality of bundles via bordism invariants that prevent global framing. This analytic computation via elliptic operators confirms topological obstructions without direct bundle .

Examples

Compact Examples

Compact parallelizable manifolds include all compact , which are parallelizable because their provides a basis for left-invariant vector fields that are globally defined and linearly independent everywhere on the manifold. These vector fields form a trivialization of the , as the left translations ensure a consistent framing across the group. The nn-dimensional tori Tn=Rn/ZnT^n = \mathbb{R}^n / \mathbb{Z}^n provide another class of compact parallelizable manifolds, which can be viewed as abelian compact groups. They admit a global frame consisting of constant vector fields aligned with the coordinate directions ∂/∂θi\partial / \partial \theta_i, i=1,…,ni=1,\dots,n, in toroidal coordinates, which are everywhere linearly independent and trivially due to the flat metric. By Stiefel's theorem, every compact orientable 3-manifold is parallelizable. This includes examples like the 3-sphere S3S^3 and the Poincaré homology sphere, where the tangent bundle admits a global trivialization, often constructed using the manifold's framing properties in low dimensions. Lens spaces provide further examples of compact parallelizable 3-manifolds. Specifically, the lens spaces L(7,1)L(7,1) and L(7,2)L(7,2) are homotopy equivalent but not homeomorphic, yet the total spaces of their tangent bundles are diffeomorphic. Among the nn-spheres, only S1S^1, S3S^3, and S7S^7 are parallelizable. The $3−sphere-sphere S^3,diffeomorphictothecompactLiegroup, diffeomorphic to the compact Lie group \mathrm{SU}(2)ofunitquaternions,inheritsparallelizabilityfromitsgroupstructurevialeft−invariantvectorfields.[](https://www2.math.upenn.edu/ wziller/math650/LieGroupsReps.pdf)Anexplicitparallelizationofof unit quaternions, inherits parallelizability from its group structure via left-invariant vector fields.[](https://www2.math.upenn.edu/~wziller/math650/LieGroupsReps.pdf) An explicit parallelization ofS^3canbeconstructedusingtheHopffibrationcan be constructed using the Hopf fibrationS^1 \to S^3 \to S^2,whichfoliates, which foliates S^3intocircles;onevectorfieldinto circles; one vector fieldTistheunittangentalongthesefibers(aReeb−likefieldfromquaternionmultiplication),whiletheothertwois the unit tangent along these fibers (a Reeb-like field from quaternion multiplication), while the other twoT_2, T_3spanthehorizontaldistributionorthogonaltothefibers,extendedviathealmostcomplexstructureonspan the horizontal distribution orthogonal to the fibers, extended via the almost complex structure onS^2$. The $7−sphere-sphere S^7,identifiedwiththeunitoctonions,admitsaparallelizationinducedbyrightmultiplicationbytheimaginaryoctonionicbasis, identified with the unit octonions, admits a parallelization induced by right multiplication by the imaginary octonionic basis {e_1, \dots, e_7},yieldingsevenvectorfields, yielding seven vector fields X_i(x) = e_i xforforx \in S^7$, which are orthonormal and globally defined with respect to the standard metric. These fields trivialize the tangent bundle, though their Lie brackets reflect the non-associativity of octonions. Exotic spheres also illustrate properties related to parallelizable manifolds. For instance, exotic 7-spheres are smooth manifolds homeomorphic but not diffeomorphic to the standard S7S^7. The total space of the tangent bundle of any exotic nn-sphere is diffeomorphic to that of the standard SnS^n.

Example of Non-Triviality: The 2-Sphere

The 2-sphere S2S^2 serves as a compact example of a non-parallelizable manifold, illustrating the non-triviality of its tangent bundle TS2TS^2. Specifically, the total space of TS2TS^2 is not homeomorphic to that of the trivial bundle S2×R2S^2 \times \mathbb{R}^2. To demonstrate this non-homeomorphism, first consider the unit tangent bundle US2⊂TS2US^2 \subset TS^2 (the set of all unit-length tangent vectors) and its analogue S1×S2⊂R2×S2S^1 \times S^2 \subset \mathbb{R}^2 \times S^2. The action of SO(3)SO(3) on S2S^2 extends to US2US^2, and this action is transitive with trivial stabilizer, inducing a diffeomorphism SO(3)→US2SO(3) \to US^2. Consequently, π1(US2)≅π1(SO(3))≅Z/2Z.\pi_1(US^2) \cong \pi_1(SO(3)) \cong \mathbb{Z}/2\mathbb{Z}. In contrast, π1(S1×S2)≅Z\pi_1(S^1 \times S^2) \cong \mathbb{Z}, so US2US^2 is not homeomorphic to S1×S2S^1 \times S^2. Now, let X=TS2X = TS^2 and Y=R2×S2Y = \mathbb{R}^2 \times S^2, and denote their one-point compactification by X∪{∞}X \cup \{\infty\} and Y∪{∞}Y \cup \{\infty\}. The relative homology groups satisfy H2(X∪{∞},X)≅H1(US2)≅Z/2ZH_2(X \cup \{\infty\}, X) \cong H_1(US^2) \cong \mathbb{Z}/2\mathbb{Z} and H2(Y∪{∞},Y)≅H1(S1×S2)≅Z.H_2(Y \cup \{\infty\}, Y) \cong H_1(S^1 \times S^2) \cong \mathbb{Z}. To establish the first isomorphism, let S⊂TS2S \subset TS^2 be the zero section (the canonical copy of S2S^2). By excision, H2(X∪{∞},X)≅H2(X∪{∞}−S,X−S).H_2(X \cup \{\infty\}, X) \cong H_2(X \cup \{\infty\} - S, X - S). The space X∪{∞}−SX \cup \{\infty\} - S is contractible, so by the long exact sequence of the pair (X∪{∞}−S,X−S)(X \cup \{\infty\} - S, X - S), H2(X∪{∞}−S,X−S)≅H1(X−S).H_2(X \cup \{\infty\} - S, X - S) \cong H_1(X - S). Moreover, X−SX - S deformation retracts onto US2US^2, hence H1(X−S)≅H1(US2)H_1(X - S) \cong H_1(US^2). A similar argument establishes the second isomorphism. Since the relative homology groups differ, the one-point compactifications are not homeomorphic, implying that XX and YY are not homeomorphic.

Non-Compact Examples

The Euclidean space Rn\mathbb{R}^n exemplifies a non-compact parallelizable manifold, with its tangent bundle trivialized by the global coordinate frame {∂∂x1,…,∂∂xn}\left\{ \frac{\partial}{\partial x_1}, \dots, \frac{\partial}{\partial x_n} \right\}. More broadly, any contractible non-compact manifold is parallelizable, as its tangent bundle, being a vector bundle over a contractible base, is trivial. This includes open subsets of Rn\mathbb{R}^n, such as open balls, and contractible non-compact surfaces like the Euclidean plane R2\mathbb{R}^2. The hyperbolic space HnH^n provides another fundamental non-compact example, diffeomorphic to the open Euclidean ball Bn⊂RnB^n \subset \mathbb{R}^n via models like the Poincaré ball, thereby inheriting the trivial tangent bundle of Rn\mathbb{R}^n. Non-compact solvmanifolds and nilmanifolds, as quotients of solvable or groups by discrete subgroups acting freely by left translations, are parallelizable; the left-invariant frame on the projects to a global framing on the quotient. These serve as non-compact analogs of tori, with the triviality arising from the group structure rather than compactness.

Properties and Theorems

Key Theorems

One of the foundational results in the study of parallelizable manifolds is the theorem on the parallelizability of spheres, proved by and . The theorem states that the nn-dimensional SnS^n admits a parallelization n=1,3,n = 1, 3, or $7.Thisresultreliesonthefactthataparallelizationof. This result relies on the fact that a parallelization of S^ncorrespondstoatrivializationofits[tangentbundle](/page/Tangentbundle),whichisclassifiedbytheclutchingconstructionoverthe[equator](/page/Equator)corresponds to a trivialization of its [tangent bundle](/page/Tangent_bundle), which is classified by the clutching construction over the [equator](/page/Equator)S^{n-1}.Theexistenceofsuchatrivializationisequivalenttotheclutchingmapbeingnull−homotopicinthe[homotopygroup](/page/Homotopygroup). The existence of such a trivialization is equivalent to the clutching map being null-homotopic in the [homotopy group](/page/Homotopy_group) \pi_{n-1}(SO(n)).UsingBottperiodicity,whichdescribesthestablehomotopygroupsoftheorthogonalgroupsasperiodicwithperiod8(specifically,. Using Bott periodicity, which describes the stable homotopy groups of the orthogonal groups as periodic with period 8 (specifically, \pi_k(O) \cong \mathbb{Z}_2forfork \equiv 0,1 \pmod{8},, \mathbb{Z}forfork \equiv 3,7 \pmod{8}, and $0 otherwise, with π2≅0\pi_2 \cong 0 and π4≅0\pi_4 \cong 0, π5≅0\pi_5 \cong 0, π6≅0\pi_6 \cong 0), detailed computations of the relevant unstable groups πn−1(SO(n))\pi_{n-1}(SO(n)) show that the obstruction to parallelizability vanishes precisely for n=1,3,7n=1,3,7. This theorem highlights the exceptional nature of dimensions 1, 3, and 7, linked to the existence of division algebras over the reals. A key realization, attributed to Steenrod in the context of bundle theory, is that parallelizability of a smooth manifold implies the vanishing of all its characteristic classes. Specifically, if the tangent bundle TMTM is trivial, then the Stiefel-Whitney classes wi(TM)=0w_i(TM) = 0 for all i≥1i \geq 1, the Pontryagin classes pi(TM)=0p_i(TM) = 0 for all i≥1i \geq 1, the Euler class e(TM)=0e(TM) = 0, and (for oriented or complex cases) the Chern classes ci(TM)=0c_i(TM) = 0. This follows directly from the defining properties of characteristic classes for vector bundles: the total characteristic class of a trivial bundle is 1 in the appropriate cohomology ring. For instance, the Stiefel-Whitney classes, defined via the Thom isomorphism and Steenrod squares, must be zero because the classifying map to BO(n)BO(n) factors through the trivial bundle's classifying space EO(n)EO(n), which has trivial cohomology in positive degrees. These vanishing conditions serve as necessary obstructions to parallelizability, though not always sufficient. Bott periodicity plays a central role beyond the sphere theorem, linking the parallelizability of spheres to the periodic structure of and stable homotopy. In particular, the periodicity implies that the stable tangent bundle of SnS^n is trivial only in dimensions where the real K-group KO−n(pt)≅ZKO^{-n}(pt) \cong \mathbb{Z} or Z/2Z\mathbb{Z}/2\mathbb{Z} aligns with the frameability conditions, but the precise connection for unstable parallelizability reinforces the exceptions at n=1,3,7n=1,3,7. Compact provide a canonical class of parallelizable manifolds. Every GG, compact or not, is parallelizable, with the trivialization given by left-invariant vector fields. Let g\mathfrak{g} be the of GG, and fix a basis {e1,…,en}\{e_1, \dots, e_n\} of g\mathfrak{g}. For each eie_i, define the left-invariant vector field Xi(g)=dLg(ei)X_i(g) = dL_g(e_i), where Lg:G→GL_g: G \to G is left multiplication by gg. These fields X1,…,XnX_1, \dots, X_n form a global frame for TGTG, as they are everywhere linearly independent (by the inverse function theorem at the identity) and span the tangent spaces via the group structure. For compact GG, this also implies orientability and vanishing Euler characteristic. Wang's theorem addresses parallelizability in the context of homogeneous spaces, particularly for complex structures. For compact complex parallelizable manifolds, H. C. proved that such a manifold MM is diffeomorphic to G/ΓG/\Gamma, where GG is a complex and Γ\Gamma is a discrete central subgroup acting freely and properly. The proof proceeds by showing that the holomorphic tangent bundle being trivial implies MM admits a transitive action by a complex , with the structure group reducing to the center, leading to the quotient form. This extends to real homogeneous spaces where the isotropy representation allows trivialization.

Invariants and Obstructions

A primary obstruction to the parallelizability of an even-dimensional orientable manifold MM is the non-vanishing of its e(TM)∈H2k(M;Z)e(TM) \in H^{2k}(M; \mathbb{Z}), where dim⁡M=2k\dim M = 2k. If e(TM)≠0e(TM) \neq 0, then MM admits no nowhere-vanishing , precluding the existence of a full framing of the TMTM. This obstruction arises as the primary class detecting the non-triviality of the sphere bundle of TMTM. Secondary obstructions to parallelizability lie in higher cohomology groups and involve more refined invariants, such as Massey products, which capture interactions among primary obstructions in the Postnikov tower of the for oriented frame bundles. These higher-order invariants, valued in cohomology with coefficients in homotopy groups of the special orthogonal group SO(n)SO(n), determine whether an initial partial framing can be extended over the entire manifold. The Atiyah-Hirzebruch provides a tool for computing these obstruction groups by converging to the cohomology of the manifold with coefficients in the homotopy groups of the Vn(R)V_n(\mathbb{R}), the for nn-framings. The E2E_2-page of this sequence is given by E2p,q=Hp(M;πq(Vn(R)))E_2^{p,q} = H^p(M; \pi_q(V_n(\mathbb{R}))), with differentials encoding higher relations among characteristic classes and homotopy data. Parallelizability is closely related to stable parallelizability, where TM⊕ϵ1TM \oplus \epsilon^1 is trivial but TMTM itself is not; a classic example is the 2-sphere S2S^2, whose tangent bundle has Euler class e(TS2)=2[S2]≠0e(TS^2) = 2[S^2] \neq 0, obstructing parallelizability, yet TS2⊕ϵ1TS^2 \oplus \epsilon^1 is trivial due to the triviality of the normal bundle in R3\mathbb{R}^3. In dimension 3, every compact 3-manifold is parallelizable, as established by Stiefel's theorem, which follows from the vanishing of the relevant Stiefel-Whitney classes w1(TM)=0w_1(TM) = 0 and w2(TM)=0w_2(TM) = 0 for orientability and the absence of further obstructions in low dimensions. This result highlights how dimensional constraints can eliminate all obstructions to triviality of the .

Applications

In Lie Theory

In Lie theory, a fundamental result is that every , whether compact or non-compact, admits a trivial , making it parallelizable. This follows from the existence of global left-invariant (or right-invariant) vector fields, which provide a nowhere-vanishing frame for the at every point. Specifically, if GG is a with g=TeG\mathfrak{g} = T_e G, then for any basis {e1,…,en}\{e_1, \dots, e_n\} of g\mathfrak{g}, the left-invariant vector fields Xi(g)=dLg(ei)X_i(g) = dL_g(e_i), where LgL_g denotes left multiplication by g∈Gg \in G, form a global basis for TGTG. The map Ψ:G×g→TG\Psi: G \times \mathfrak{g} \to TG given by Ψ(g,v)=dLg(v)\Psi(g, v) = dL_g(v) is then a , confirming the triviality of TGTG. This parallelizability extends to homogeneous spaces G/HG/H, where GG is a and HH is a closed , under certain conditions on the action of HH on the . The of G/HG/H is the G×H(g/h)G \times_H (\mathfrak{g}/\mathfrak{h}), where h\mathfrak{h} is the of HH. For G/HG/H to be parallelizable, there must exist an HH-invariant basis of the at the base point o=eHo = eH, which is identified with the HH-module m=g/hm = \mathfrak{g}/\mathfrak{h} via the representation Ad∣H:H→GL(m)\mathrm{Ad}|_H: H \to \mathrm{GL}(m). If such an invariant frame exists on mm, it extends via the transitive GG-action to a global parallel frame on G/HG/H, trivializing the . In the special case where HH acts trivially on mm, any basis serves as invariant, ensuring parallelizability; more generally, the existence depends on the representation admitting an HH-equivariant trivialization. For instance, when H={e}H = \{e\}, G/H=GG/H = G recovers the case. Parallelizability in the context of principal bundles over groups ties directly to the triviality of frame bundles. A manifold MM is parallelizable if and only if its P→MP \to M, a principal GL(n,R)\mathrm{GL}(n,\mathbb{R})-bundle, is trivial as a . For a principal GG-bundle π:P→B\pi: P \to B with structure group GG a , the total space PP inherits parallelizability from GG via the vertical subbundle, which is trivialized by left-invariant fields on each fiber diffeomorphic to GG. However, the base BB is parallelizable only if the full of BB (or equivalently, the associated ) admits a global section, often requiring the bundle to be trivial or the connections to allow flat reductions in the structure. This relation underscores how Lie-theoretic constructions, such as reductions of structure groups, facilitate trivializations in bundle . The exponential map further highlights parallelizability by providing coordinate charts that exploit the structure. For a general GG, the exponential map exp⁡:g→G\exp: \mathfrak{g} \to G is a near the origin, yielding in a neighborhood of the identity, where left-invariant fields align with coordinate vector fields. In special cases, such as simply connected nilpotent groups, exp⁡\exp is a global , endowing GG with global coordinates isomorphic to Rn\mathbb{R}^n via g\mathfrak{g}, directly reflecting the trivial . Even for compact groups, where exp⁡\exp is surjective but not injective, it parametrizes geodesics and aids in constructing invariant frames. A notable distinction arises in examples like SO(3)\mathrm{SO}(3) and SU(2)\mathrm{SU}(2), both compact groups hence parallelizable, but illustrating different topological features within . SU(2)\mathrm{SU}(2) is diffeomorphic to S3S^3, simply connected, with its su(2)\mathfrak{su}(2) yielding left-invariant frames that trivialize TSU(2)T\mathrm{SU}(2). In contrast, SO(3)≅SU(2)/{±I}\mathrm{SO}(3) \cong \mathrm{SU}(2)/\{\pm I\} is the by , non-simply connected, yet remains parallelizable as the discrete action preserves the trivial . This pair exemplifies how parallelizability holds uniformly for groups regardless of connectivity, while the covering relationship affects other invariants like the .

In Topology and Geometry

In , parallelizable manifolds play a crucial role by simplifying the process of handlebody decompositions and the attachment of during the construction of cobordisms between manifolds. Since the of a parallelizable manifold is trivial, it admits a global framing, which eliminates the need for additional stable trivializations when performing surgeries along embedded spheres; this triviality ensures that the normal bundle to the is stably trivial, facilitating the computation of the surgery obstruction groups and the up to . For instance, in the study of simply-connected manifolds, this property allows for straightforward handle attachments without framing anomalies, as detailed in foundational works on the subject. Parallelizable manifolds contribute significantly to computations in oriented cobordism groups, where their trivial tangent bundles enable the construction of Thom classes and orientations in cohomology theories, thereby representing specific bordism classes that aid in determining the structure of these groups. In oriented cobordism, stably parallelizable manifolds, including parallelizable ones, are h*-orientable for any generalized cohomology theory h*, allowing their classes to generate or bound elements in the cobordism ring ΩSO_n; this property simplifies calculations by relating oriented bordism to stable homotopy groups via the Pontryagin-Thom construction. For example, the oriented cobordism group in low dimensions often involves parallelizable examples like tori or spheres, whose triviality helps resolve the additive structure of ΩSO_*. Parallelizable manifolds admit codimension-zero foliations in a trivial manner, as the entire manifold serves as a single leaf, with the trivial providing a global parallel frame for the 's tangent distribution. This trivial highlights the inherent parallelism of the manifold's , distinguishing it from non-parallelizable cases where even the full-dimensional lacks a consistent framing. In metric , parallelizable manifolds such as tori support constant metrics, exemplified by the flat Euclidean metric on the n-torus T^n, which induces zero everywhere due to the trivial allowing a global coordinate frame compatible with the flat structure. This metric arises from identifying opposite faces of a or , yielding a Riemannian metric of constant zero that is complete and invariant under the torus's action. Such constructions underscore how parallelizability facilitates the existence of homogeneous metrics on these spaces. Exotic 7-spheres, as constructed by Milnor, are parallelizable, inheriting the trivial property of the standard 7-sphere through their equivalence and stable parallelizability. Milnor demonstrated the of 28 distinct smooth structures on the 7-sphere by analyzing bundles over S^4 with S^3, and subsequent results confirm that all such exotic spheres admit a global framing of linearly independent vector fields, as their bundles are isomorphic to that of S^7 via equivalences. This parallelizability follows from the fact that 7-spheres are stably parallelizable, and the maximum number of independent vector fields matches that of S^7, equaling 7.

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