6.2 Fundamental class

The fundamental class is a cornerstone of algebraic topology, assigning a special homology class to oriented manifolds. It captures essential topological properties and serves as a generator for certain homology groups, providing insights into the relationship between topology and geometry.

Understanding the fundamental class is crucial for exploring Poincaré duality, intersection theory, and various applications in mathematics and physics. It connects homology and cohomology, allowing for powerful computations and revealing deep symmetries within manifolds.

Definition of fundamental class

• The fundamental class is a key concept in algebraic topology that assigns a special homology class to oriented manifolds
• It captures the essential topological properties of the manifold and serves as a generator for certain homology groups
• Understanding the fundamental class is crucial for studying the relationship between the topology and geometry of manifolds

Fundamental class for manifolds

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• For an oriented closed manifold $M$ of dimension $n$, the fundamental class $[M]$ is a generator of the top homology group $H_n(M; \mathbb{Z})$
• The choice of orientation determines the sign of the fundamental class, with opposite orientations corresponding to negatives of each other
• The fundamental class represents the entire manifold as a single homology class, encoding its global topological structure

Fundamental class in homology

• The fundamental class lives in the homology group of the manifold, which is an algebraic invariant capturing the "holes" and connectivity of the space
• In singular homology, the fundamental class can be represented by a sum of oriented simplices that triangulate the manifold
• The fundamental class is a homology class with integer coefficients, reflecting the orientability of the manifold

Orientation and fundamental class

• The existence of a fundamental class relies on the manifold being orientable, meaning it admits a consistent choice of orientation for all its tangent spaces
• Orientability is a global property that allows for a coherent notion of "positive" and "negative" directions on the manifold
• Non-orientable manifolds, such as the Möbius strip or Klein bottle, do not possess a fundamental class

Properties of fundamental class

• The fundamental class exhibits several important properties that make it a powerful tool in algebraic topology
• These properties reflect the intrinsic nature of the manifold and its relationship with other topological invariants
• Understanding these properties is essential for applying the fundamental class in various contexts

Uniqueness of fundamental class

• For a given oriented closed manifold, the fundamental class is unique up to sign
• Any two fundamental classes of the same manifold differ only by a factor of $\pm 1$, corresponding to the choice of orientation
• This uniqueness property ensures that the fundamental class is a well-defined invariant of the manifold

Invariance under homotopy equivalence

• The fundamental class is preserved under homotopy equivalence, meaning that homotopy equivalent manifolds have the same fundamental class (up to sign)
• Homotopy equivalence is a weaker notion than homeomorphism, allowing for continuous deformations that may not be invertible
• This invariance property makes the fundamental class a topological invariant, independent of the specific geometry of the manifold

Naturality of fundamental class

• The fundamental class behaves naturally with respect to continuous maps between oriented manifolds
• Given a continuous map $f: M \to N$ between oriented closed manifolds, the induced map on homology $f_*: H_n(M) \to H_n(N)$ satisfies $f_*([M]) = \deg(f) [N]$, where $\deg(f)$ is the degree of the map
• This naturality property allows for the study of the fundamental class under mappings and provides a connection between the topology of different manifolds

Fundamental class and Poincaré duality

• Poincaré duality is a profound result in algebraic topology that relates the homology and cohomology groups of a manifold
• The fundamental class plays a central role in the formulation and proof of Poincaré duality
• Understanding the relationship between the fundamental class and Poincaré duality is crucial for exploring the deeper structure of manifolds

Statement of Poincaré duality

• For an oriented closed manifold $M$ of dimension $n$, Poincaré duality asserts an isomorphism between the homology group $H_k(M)$ and the cohomology group $H^{n-k}(M)$
• This isomorphism is given by the cap product with the fundamental class: $\cap [M]: H^{n-k}(M) \to H_k(M)$
• Poincaré duality reveals a deep symmetry between homology and cohomology, allowing for the study of one in terms of the other

Fundamental class as generator

• The fundamental class serves as a generator for the top homology group $H_n(M)$, which is isomorphic to $\mathbb{Z}$ for oriented closed manifolds
• Under the Poincaré duality isomorphism, the fundamental class corresponds to the generator of the zeroth cohomology group $H^0(M)$
• This dual role of the fundamental class as a generator in both homology and cohomology highlights its importance in the study of manifolds

Cap product with fundamental class

• The cap product is a bilinear pairing between cohomology and homology, defined as $\cap: H^k(M) \times H_n(M) \to H_{n-k}(M)$
• In the context of Poincaré duality, the cap product with the fundamental class induces the isomorphism between homology and cohomology
• The cap product with the fundamental class can be used to compute homology classes from cohomology classes and vice versa

Applications of fundamental class

• The fundamental class finds numerous applications in various areas of mathematics, including geometry, topology, and physics
• These applications demonstrate the power and versatility of the fundamental class in solving problems and uncovering connections between different fields
• Exploring these applications helps to appreciate the significance of the fundamental class beyond its purely algebraic definition

Intersection theory using fundamental class

• Intersection theory studies the intersection of submanifolds within a larger manifold, and the fundamental class plays a key role in this context
• The intersection of two submanifolds can be computed by taking the cap product of their Poincaré dual cohomology classes with the fundamental class of the ambient manifold
• The fundamental class allows for the computation of intersection numbers, which provide important geometric and topological information about the submanifolds

Degree of map and fundamental class

• The degree of a continuous map $f: M \to N$ between oriented closed manifolds of the same dimension can be defined using the fundamental class
• The degree of $f$ is the integer $\deg(f)$ such that $f_*([M]) = \deg(f) [N]$, where $f_*$ is the induced map on homology
• The degree of a map provides information about the covering properties and the behavior of the map on the level of homology

Euler characteristic via fundamental class

• The Euler characteristic is a topological invariant that measures the "shape" of a manifold, taking into account its vertices, edges, and faces
• For an oriented closed manifold $M$ of even dimension $n$, the Euler characteristic can be computed as the self-intersection number of the fundamental class: $\chi(M) = [M] \cdot [M]$
• This connection between the Euler characteristic and the fundamental class provides a way to compute the Euler characteristic using algebraic topology

Computations with fundamental class

• Computing the fundamental class explicitly for specific manifolds is an important task in algebraic topology
• These computations often involve techniques from homology theory and the properties of the manifolds under consideration
• By calculating the fundamental class for well-known manifolds, one can gain insights into their topological structure and apply them to more general cases

Fundamental class of spheres

• The $n$-dimensional sphere $S^n$ is a fundamental example of a closed orientable manifold, and its fundamental class is well-understood
• For the sphere $S^n$, the fundamental class $[S^n]$ is a generator of the top homology group $H_n(S^n) \cong \mathbb{Z}$
• The choice of orientation for the sphere determines the sign of the fundamental class, with the standard orientation corresponding to the positive generator

Fundamental class of projective spaces

• Projective spaces, such as the real projective space $\mathbb{RP}^n$ and the complex projective space $\mathbb{CP}^n$, are important examples of manifolds with rich topological structure
• The fundamental class of $\mathbb{RP}^n$ is a generator of $H_n(\mathbb{RP}^n; \mathbb{Z}/2\mathbb{Z}) \cong \mathbb{Z}/2\mathbb{Z}$, reflecting the non-orientability of real projective spaces
• For complex projective spaces, the fundamental class of $\mathbb{CP}^n$ generates $H_{2n}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}$, capturing the complex structure of the manifold

Fundamental class of product manifolds

• The fundamental class of a product manifold can be expressed in terms of the fundamental classes of its factors
• For oriented closed manifolds $M$ and $N$, the fundamental class of the product manifold $M \times N$ is given by the cross product of the fundamental classes: $[M \times N] = [M] \times [N]$
• This property allows for the computation of the fundamental class of product manifolds using the fundamental classes of simpler manifolds

Generalizations of fundamental class

• The concept of the fundamental class can be generalized and extended to various settings beyond oriented closed manifolds
• These generalizations allow for the study of more diverse topological spaces and provide a broader framework for understanding the role of the fundamental class
• Exploring these generalizations helps to appreciate the versatility and adaptability of the fundamental class in different contexts

Fundamental class in cohomology

• The fundamental class can also be defined in cohomology, where it lives in the top cohomology group of the manifold
• For an oriented closed manifold $M$ of dimension $n$, the fundamental class in cohomology is an element $[M]^* \in H^n(M; \mathbb{Z})$ that pairs with the fundamental class in homology to give the orientation
• The fundamental class in cohomology is Poincaré dual to the fundamental class in homology and plays a similar role in the study of manifolds

Fundamental class for orbifolds

• Orbifolds are generalizations of manifolds that allow for certain types of singularities, such as quotient singularities arising from group actions
• The concept of the fundamental class can be extended to orbifolds, taking into account the presence of singularities and the orbifold structure
• The fundamental class of an orbifold captures the topological and geometric information of the underlying space, including the contribution from the singular points

Virtual fundamental class

• In some situations, such as in the study of moduli spaces or in the presence of obstructions, the fundamental class may not exist in the usual sense
• The virtual fundamental class is a generalization that assigns a homology class to certain spaces, even when the actual fundamental class is not well-defined
• Virtual fundamental classes play a crucial role in enumerative geometry and the study of invariants associated with moduli spaces, such as Gromov-Witten invariants