6.1 Orientation of manifolds

Orientation of manifolds is a crucial concept in topology, determining how a manifold's local structure aligns globally. It's key for defining integrals, understanding homology and cohomology, and distinguishing between orientable and non-orientable surfaces.

This topic connects to broader themes in cohomology theory by exploring how orientation affects integration, Poincaré duality, and characteristic classes. It also links to intersection theory and degree theory, showcasing the far-reaching implications of a manifold's orientability.

Orientation of manifolds

• Fundamental concept in the study of manifolds and their topological properties
• Plays a crucial role in understanding the behavior of integrals, homology, and cohomology on manifolds
• Closely related to the notions of orientability, which determines whether a consistent choice of orientation is possible on a manifold

Definition of orientation

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• Orientation is a choice of a consistent local ordering or "handedness" on a manifold
• For an $n$-dimensional manifold, an orientation is a continuous choice of an ordered basis for the tangent space at each point
• Equivalently, an orientation is a continuous choice of a non-vanishing $n$-form (volume form) on the manifold
• The two possible orientations on a connected orientable manifold are called "positive" and "negative" orientations

Orientable vs non-orientable manifolds

• A manifold is called orientable if it admits a consistent choice of orientation
• Examples of orientable manifolds include the sphere, torus, and all surfaces of even genus (sphere, torus, double torus, etc.)
• Non-orientable manifolds do not admit a consistent choice of orientation
• Examples of non-orientable manifolds include the Möbius strip, Klein bottle, and all surfaces of odd genus (projective plane, Klein surface, etc.)

Local vs global orientation

• Local orientation refers to the choice of orientation on a small neighborhood of a point on a manifold
• Global orientation refers to a consistent choice of local orientations across the entire manifold
• A manifold is orientable if and only if it admits a global orientation
• The existence of a global orientation is equivalent to the triviality of the orientation double cover of the manifold

Orientation-preserving maps

• A continuous map between oriented manifolds is called orientation-preserving if it preserves the chosen orientations
• Formally, a map $f: M \to N$ between oriented manifolds is orientation-preserving if the pullback of the orientation form on $N$ equals the orientation form on $M$
• Orientation-preserving maps play a crucial role in the study of degree theory and the behavior of integrals under mappings
• Examples of orientation-preserving maps include rotations and translations of Euclidean space

Orientation and homology

• Orientation plays a crucial role in the definition and properties of homology groups
• For an oriented $n$-dimensional manifold $M$, the top homology group $H_n(M; \mathbb{Z})$ is isomorphic to $\mathbb{Z}$, with a generator corresponding to the fundamental class $[M]$ determined by the orientation
• The choice of orientation determines the sign of the fundamental class and the induced orientations on the boundary of chains
• Poincaré duality relates the homology and cohomology of an oriented manifold via the cap product with the fundamental class

Orientation and cohomology

• Orientation is essential in the definition and properties of cohomology groups
• For an oriented $n$-dimensional manifold $M$, the top cohomology group $H^n(M; \mathbb{Z})$ is isomorphic to $\mathbb{Z}$, with a generator corresponding to the orientation class $[M]^*$ dual to the fundamental class
• The cup product in cohomology is graded-commutative for oriented manifolds, with the sign determined by the orientation
• The orientation class plays a crucial role in the formulation of Poincaré duality and the definition of the Euler class of vector bundles

Orientation and integration

• Orientation is necessary for defining integration on manifolds
• An orientation on a manifold determines a choice of a volume form, which is used to define the integral of differential forms
• The change of variables formula for integrals involves the Jacobian determinant, which depends on the chosen orientation
• Stokes' theorem relates the integral of a differential form over a manifold to the integral of its exterior derivative over the boundary, with signs determined by the induced orientation on the boundary

Orientation and Poincaré duality

• Poincaré duality is a fundamental result relating homology and cohomology of oriented manifolds
• For an oriented $n$-dimensional closed manifold $M$, Poincaré duality states that the cap product with the fundamental class $[M]$ induces isomorphisms $H^k(M; \mathbb{Z}) \cong H_{n-k}(M; \mathbb{Z})$ for all $k$
• The orientation of $M$ determines the sign of the fundamental class and the induced isomorphisms
• Poincaré duality has numerous applications, including the study of intersection pairings, the Lefschetz fixed-point theorem, and the Hodge theorem

Orientation of boundaries

• The boundary of an oriented manifold inherits a natural orientation
• For an oriented $n$-dimensional manifold $M$ with boundary $\partial M$, the induced orientation on $\partial M$ is determined by the "outward normal first" convention
• The boundary orientation is crucial in the formulation of Stokes' theorem and the behavior of integrals over manifolds with boundary
• The relationship between the orientation of a manifold and its boundary is captured by the long exact sequence in homology and cohomology

Orientation of products

• The product of oriented manifolds inherits a natural orientation
• For oriented manifolds $M$ and $N$, the product orientation on $M \times N$ is defined by the tensor product of the orientation forms on $M$ and $N$
• The Künneth theorem relates the homology and cohomology of a product manifold to the homology and cohomology of its factors, with signs determined by the product orientation
• The cross product in cohomology is defined using the product orientation and satisfies the graded-commutativity property

Orientation and covering spaces

• The orientation of a manifold lifts to its covering spaces
• For an oriented manifold $M$ and a covering space $p: \tilde{M} \to M$, the orientation of $M$ lifts to a unique orientation on $\tilde{M}$ such that $p$ is orientation-preserving
• The orientation double cover of a non-orientable manifold is the minimal orientable covering space
• The orientation behavior under covering maps is related to the deck transformation group and the monodromy action on fibers

Orientation and vector bundles

• Orientation of vector bundles is a generalization of the orientation of manifolds
• A vector bundle is called orientable if it admits a consistent choice of orientation for each fiber
• The orientation of a vector bundle is equivalent to a reduction of its structure group to $SO(n)$
• The Euler class of an oriented vector bundle is a characteristic class that measures the twisting of the orientation and is related to the Euler characteristic of the base manifold

Orientation and characteristic classes

• Characteristic classes are cohomological invariants associated with vector bundles and provide information about their global topological properties
• The orientation of a vector bundle plays a crucial role in the definition and properties of characteristic classes
• The Euler class is a characteristic class defined for oriented vector bundles and is related to the obstruction to the existence of non-vanishing sections
• The Pontryagin classes are characteristic classes defined for real vector bundles and are related to the obstruction to the existence of complex structures

Orientation and intersection theory

• Intersection theory studies the intersection properties of submanifolds and the resulting topological invariants
• Orientation is essential in defining the intersection product and the signed count of intersection points
• The intersection pairing between homology classes of complementary dimensions is well-defined for oriented manifolds
• The intersection form of a 4-manifold is a powerful invariant that captures the intersection properties of 2-dimensional homology classes and is related to the signature and the Donaldson invariants

Orientation and degree theory

• Degree theory studies the behavior of continuous maps between oriented manifolds of the same dimension
• The degree of a map is an integer-valued homotopy invariant that counts the number of preimages of a regular value with signs determined by the orientation
• The degree is a powerful tool for studying the existence and multiplicity of solutions to nonlinear equations
• The Brouwer fixed-point theorem and the Borsuk-Ulam theorem are classic applications of degree theory

Orientation and Euler characteristic

• The Euler characteristic is a topological invariant that measures the alternating sum of the Betti numbers of a manifold
• For closed oriented surfaces, the Euler characteristic is related to the genus by the formula $\chi = 2 - 2g$
• The Euler characteristic satisfies the product formula $\chi(M \times N) = \chi(M) \cdot \chi(N)$ for oriented manifolds
• The Gauss-Bonnet theorem relates the Euler characteristic of a closed oriented Riemannian 2-manifold to its total curvature

Orientation and Stiefel-Whitney classes

• Stiefel-Whitney classes are characteristic classes associated with real vector bundles and provide information about their orientability and topology
• The first Stiefel-Whitney class $w_1$ is the obstruction to the orientability of a vector bundle
• The second Stiefel-Whitney class $w_2$ is the obstruction to the existence of a spin structure on a manifold
• The Whitney product formula relates the Stiefel-Whitney classes of a tensor product of vector bundles to those of the factors

Orientation and spin structures

• A spin structure on an oriented Riemannian manifold is a lift of the orthonormal frame bundle to a principal $\text{Spin}(n)$-bundle
• The existence of a spin structure is equivalent to the vanishing of the second Stiefel-Whitney class $w_2$
• Spin structures are crucial in the study of spinors, Dirac operators, and the Atiyah-Singer index theorem
• The cobordism theory of spin manifolds is a powerful tool in the classification of manifolds and the study of their invariants

Orientation and surgery theory

• Surgery theory is a technique for modifying manifolds by cutting and gluing along embedded spheres or disks
• Orientation plays a crucial role in the definition of the surgery obstructions and the resulting cobordism groups
• The oriented cobordism group $\Omega^{SO}_n$ classifies oriented $n$-manifolds up to oriented cobordism
• The Wall groups $L_n(\pi)$ measure the obstruction to performing surgery on an oriented manifold with fundamental group $\pi$ to obtain a homotopy sphere

Orientation and cobordism theory

• Cobordism theory studies the equivalence classes of manifolds up to cobordism, which is a relation that captures the notion of "bounding a manifold with boundary"
• Oriented cobordism is a refinement of cobordism theory that takes into account the orientation of manifolds
• The oriented cobordism group $\Omega^{SO}_n$ is a powerful invariant that captures the oriented cobordism classes of $n$-manifolds
• The Pontryagin-Thom construction relates the oriented cobordism groups to the stable homotopy groups of spheres and provides a geometric interpretation of the latter