🧬Cohomology Theory Unit 6 – Poincaré duality

Key Concepts and Definitions

• Poincaré duality establishes a fundamental relationship between homology and cohomology groups of a manifold
• Relates the k-th homology group of a closed, oriented n-dimensional manifold M to its (n-k)-th cohomology group
  • Specifically, $H_k(M) \cong H^{n-k}(M)$, where $\cong$ denotes isomorphism
• Homology groups $H_k(M)$ capture information about k-dimensional holes in the manifold M
  • For example, $H_0(M)$ represents connected components, $H_1(M)$ represents loops, and $H_2(M)$ represents voids
• Cohomology groups $H^k(M)$ assign algebraic structures to the manifold M, capturing global properties
• The cap product $\frown$ is a bilinear pairing between homology and cohomology classes, playing a crucial role in Poincaré duality
• The fundamental class $[M] \in H_n(M)$ is a generator of the top homology group, representing the orientation of the manifold
• The Poincaré dual of a submanifold $N \subset M$ is a cohomology class $\eta \in H^{n-k}(M)$ satisfying $\eta \frown [M] = [N]$

Historical Context and Development

• Henri Poincaré introduced the concept of duality in his 1895 paper "Analysis Situs"
  • Poincaré's work laid the foundation for the development of algebraic topology
• In the 1930s, Hassler Whitney and Eduard Čech independently formulated the modern version of Poincaré duality using cohomology
• Whitney's approach relied on the cup product and the cap product, establishing the duality between homology and cohomology
• Čech's approach utilized the dual cell decomposition of a manifold, relating the homology of the original complex to the cohomology of the dual complex
• In the 1940s, Samuel Eilenberg and Norman Steenrod axiomatized homology and cohomology theories, providing a unified framework for Poincaré duality
• Further generalizations and applications of Poincaré duality were developed by mathematicians such as René Thom, John Milnor, and Friedrich Hirzebruch
  • These advancements led to the development of characteristic classes and the Atiyah-Singer index theorem

Fundamental Principles of Poincaré Duality

• Poincaré duality states that for a closed, oriented n-dimensional manifold M, there is an isomorphism between homology and cohomology groups:
  • $H_k(M) \cong H^{n-k}(M)$ for all $0 \leq k \leq n$
• The isomorphism is given by the cap product with the fundamental class $[M] \in H_n(M)$:
  • $D: H_k(M) \to H^{n-k}(M)$, defined by $D(\alpha) = \alpha \frown [M]$
• The Poincaré dual of a closed, oriented submanifold $N \subset M$ of codimension $k$ is a cohomology class $\eta \in H^k(M)$ satisfying:
  • $\eta \frown [M] = [N]$, where $[N] \in H_{n-k}(M)$ is the fundamental class of N
• Poincaré duality relates the intersection of submanifolds to the cup product of their Poincaré duals:
  • If $N_1, N_2 \subset M$ are transversely intersecting submanifolds with Poincaré duals $\eta_1, \eta_2$, then $\eta_1 \smile \eta_2$ is the Poincaré dual of $N_1 \cap N_2$
• The Poincaré duality isomorphism is natural with respect to continuous maps between manifolds
  • If $f: M \to N$ is a continuous map between closed, oriented manifolds, then $f_* \circ D_M = D_N \circ f^*$, where $f_*$ and $f^*$ are the induced maps on homology and cohomology, respectively

Manifolds and Orientation

• A manifold is a topological space that locally resembles Euclidean space near each point
  • More precisely, each point in an n-dimensional manifold has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$
• Manifolds can be compact (closed and bounded) or non-compact (open or unbounded)
  • Examples of compact manifolds include the circle $S^1$, the torus $T^2$, and the sphere $S^n$
  • Examples of non-compact manifolds include the real line $\mathbb{R}$, the plane $\mathbb{R}^2$, and the cylinder $S^1 \times \mathbb{R}$
• An orientation of a manifold is a consistent choice of local orientations for all points
  • In other words, it is a continuous choice of ordered bases for the tangent spaces at each point
• Orientable manifolds admit exactly two orientations, which are opposite to each other
  • Examples of orientable manifolds include the circle $S^1$, the torus $T^2$, and the sphere $S^n$
• Non-orientable manifolds do not admit a consistent choice of local orientations
  • Examples of non-orientable manifolds include the Möbius strip and the real projective plane $\mathbb{RP}^2$
• Poincaré duality holds for closed, oriented manifolds, highlighting the importance of orientability in this context

Homology and Cohomology Connections

• Homology and cohomology are dual theories that capture different aspects of the topology of a space
• Homology groups $H_k(M)$ are vector spaces that measure the presence of k-dimensional holes in a manifold M
  • Elements of $H_k(M)$ are represented by k-dimensional cycles (closed submanifolds) modulo boundaries
• Cohomology groups $H^k(M)$ are vector spaces that assign algebraic structures (such as real numbers or polynomials) to the manifold M
  • Elements of $H^k(M)$ are represented by cocycles (functions satisfying certain properties) modulo coboundaries
• The cup product $\smile$ is a bilinear operation on cohomology classes, making the direct sum of cohomology groups into a graded ring
  • The cup product is related to the intersection of submanifolds via Poincaré duality
• The cap product $\frown$ is a bilinear pairing between homology and cohomology classes, providing a link between the two theories
  • The cap product is used to define the Poincaré duality isomorphism between homology and cohomology groups
• Poincaré duality reveals a deep connection between homology and cohomology, allowing them to be used interchangeably in many contexts
  • This duality has far-reaching consequences in algebraic topology and related fields

Applications in Topology

• Poincaré duality has numerous applications in topology and related areas of mathematics
• In the study of intersection theory, Poincaré duality relates the intersection of submanifolds to the cup product of their dual cohomology classes
  • This allows for the computation of intersection numbers using algebraic methods
• Poincaré duality is a key ingredient in the formulation and proof of the Lefschetz fixed-point theorem
  • The theorem relates the fixed points of a continuous self-map of a manifold to the trace of the induced map on homology
• In the theory of characteristic classes, Poincaré duality is used to define the Chern classes of complex vector bundles and the Stiefel-Whitney classes of real vector bundles
  • These classes provide important invariants for studying the topology of vector bundles and their base spaces
• Poincaré duality plays a central role in the Atiyah-Singer index theorem, which relates the index of an elliptic differential operator on a manifold to topological invariants of the manifold
  • The index theorem has applications in differential geometry, mathematical physics, and string theory
• In the study of knots and links, Poincaré duality is used to define the linking number, which measures the entanglement of two disjoint closed curves in a 3-manifold
  • The linking number is an important invariant in knot theory and low-dimensional topology

Computational Techniques and Examples

• Computing homology and cohomology groups, as well as the Poincaré duality isomorphism, often involves algebraic and computational techniques
• Simplicial complexes and cellular complexes are combinatorial models for topological spaces that allow for the computation of homology and cohomology groups
  • The boundary operator on simplicial or cellular chains induces a differential on the corresponding cochain complex
• The homology and cohomology groups can be computed using linear algebra techniques, such as the Smith normal form algorithm for integer matrices
  • Software packages like SageMath and GAP provide tools for computing homology and cohomology of simplicial and cellular complexes
• Example: Consider the torus $T^2$, which can be represented as a cellular complex with one 0-cell, two 1-cells, and one 2-cell
  • The homology groups are $H_0(T^2) \cong \mathbb{Z}$, $H_1(T^2) \cong \mathbb{Z}^2$, and $H_2(T^2) \cong \mathbb{Z}$
  • The cohomology groups are $H^0(T^2) \cong \mathbb{Z}$, $H^1(T^2) \cong \mathbb{Z}^2$, and $H^2(T^2) \cong \mathbb{Z}$
  • Poincaré duality provides isomorphisms $H_0(T^2) \cong H^2(T^2)$, $H_1(T^2) \cong H^1(T^2)$, and $H_2(T^2) \cong H^0(T^2)$
• Example: Consider the real projective plane $\mathbb{RP}^2$, which can be represented as a cellular complex with one 0-cell, one 1-cell, and one 2-cell
  • The homology groups are $H_0(\mathbb{RP}^2) \cong \mathbb{Z}$, $H_1(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}$, and $H_2(\mathbb{RP}^2) \cong 0$
  • The cohomology groups are $H^0(\mathbb{RP}^2) \cong \mathbb{Z}$, $H^1(\mathbb{RP}^2) \cong 0$, and $H^2(\mathbb{RP}^2) \cong \mathbb{Z}/2\mathbb{Z}$
  • Poincaré duality does not apply to $\mathbb{RP}^2$ since it is a non-orientable manifold

Advanced Topics and Extensions

• Poincaré duality has been generalized and extended in various directions, leading to deeper insights and connections in algebraic topology and related fields
• The Alexander duality theorem is a generalization of Poincaré duality that relates the homology of a compact subset of a sphere to the cohomology of its complement
  • Alexander duality has applications in knot theory and the study of links in spheres
• The Poincaré-Lefschetz duality theorem is an extension of Poincaré duality to manifolds with boundary
  • It relates the homology of a manifold with boundary to the cohomology of the manifold relative to its boundary
• The Cap product and the Cup product are dual operations that can be defined more generally in the context of sheaf theory and derived categories
  • These generalizations provide a unified framework for studying duality in various contexts, such as algebraic geometry and representation theory
• Poincaré duality has analogues in other cohomology theories, such as K-theory and cobordism theory
  • These analogues relate different invariants of manifolds and have applications in topology, geometry, and mathematical physics
• In the context of algebraic geometry, the Serre duality theorem is an analogue of Poincaré duality for coherent sheaves on a smooth projective variety
  • Serre duality relates the cohomology of a sheaf to the cohomology of its dual sheaf, twisted by the canonical bundle
• The Verdier duality theorem is a generalization of Poincaré duality to the setting of sheaves on topological spaces
  • Verdier duality relates the cohomology of a sheaf to the homology of its dual sheaf, providing a powerful tool in the study of sheaf theory and derived categories