🧬Cohomology Theory Unit 3 – Relative Homology and Cohomology

Key Concepts and Definitions

• Relative homology and cohomology study topological spaces with subspaces
• Relative groups measure how the subspace sits inside the larger space
• Quotient spaces $(X,A)$ consist of a topological space $X$ and a subspace $A \subset X$
• Relative chains are chains in $X$ modulo chains in $A$, denoted $C_n(X,A) = C_n(X) / C_n(A)$
• Relative boundary operator $\partial_n: C_n(X,A) \to C_{n-1}(X,A)$ is induced by the boundary operator on $C_n(X)$
  • Satisfies $\partial_{n-1} \circ \partial_n = 0$
• Relative cycles and boundaries are defined using the relative boundary operator
• Relative homology groups $H_n(X,A)$ are the quotient of relative cycles by relative boundaries

Relative Homology Groups

• Defined for a pair of spaces $(X,A)$ where $A \subset X$
• Measure the difference between the homology of $X$ and the homology of $A$
• Computed using relative chains, cycles, and boundaries
  • Relative chains: $C_n(X,A) = C_n(X) / C_n(A)$
  • Relative cycles: $Z_n(X,A) = \ker(\partial_n: C_n(X,A) \to C_{n-1}(X,A))$
  • Relative boundaries: $B_n(X,A) = \operatorname{im}(\partial_{n+1}: C_{n+1}(X,A) \to C_n(X,A))$
• Defined as the quotient $H_n(X,A) = Z_n(X,A) / B_n(X,A)$
• Functorial with respect to maps of pairs $(X,A) \to (Y,B)$
• Related to absolute homology groups via long exact sequences

Relative Cohomology Groups

• Dual notion to relative homology groups
• Defined for a pair of spaces $(X,A)$ where $A \subset X$
• Use relative cochains $C^n(X,A) = \operatorname{Hom}(C_n(X,A), G)$ for a coefficient group $G$
• Relative coboundary operator $\delta^n: C^n(X,A) \to C^{n+1}(X,A)$ is induced by the boundary operator
  • Satisfies $\delta^{n+1} \circ \delta^n = 0$
• Relative cocycles and coboundaries are defined using the relative coboundary operator
• Relative cohomology groups $H^n(X,A;G)$ are the quotient of relative cocycles by relative coboundaries
• Functorial with respect to maps of pairs $(X,A) \to (Y,B)$ and coefficient group homomorphisms
• Related to absolute cohomology groups via long exact sequences

Long Exact Sequences

• Fundamental tool in relative homology and cohomology
• Connect relative and absolute groups in a long exact sequence
  • For homology: $\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \to \cdots$
  • For cohomology: $\cdots \to H^n(X,A;G) \to H^n(X;G) \to H^n(A;G) \to H^{n+1}(X,A;G) \to \cdots$
• Induced by short exact sequences of chain complexes or cochain complexes
• Allow for computation of relative groups from absolute groups and vice versa
• Naturality of long exact sequences with respect to maps of pairs
• Used in proving excision and other important results

Excision Theorem

• Key result in relative homology and cohomology
• States that relative groups are invariant under certain changes to the pair $(X,A)$
• Precise statement: if $U \subset A \subset X$ and the closure of $U$ is contained in the interior of $A$, then the inclusion $(X - U, A - U) \hookrightarrow (X,A)$ induces isomorphisms $H_n(X - U, A - U) \cong H_n(X,A)$ for all $n$
  • Similar statement holds for relative cohomology groups
• Allows for simplification of computations by removing "small" parts of the subspace $A$
• Proof relies on long exact sequences and homotopy invariance
• Generalizations include excision for triples and excision in other homology and cohomology theories

Applications and Examples

• Relative homology and cohomology have numerous applications in topology and geometry
• Computation of absolute groups using relative groups and long exact sequences
  • Example: homology groups of a sphere relative to a point
• Study of manifolds with boundary using relative groups of the manifold modulo its boundary
• Poincaré duality for manifolds with boundary relates absolute and relative groups
• Lefschetz duality relates the relative homology of a subspace to the absolute cohomology of its complement
• Cup and cap products in relative cohomology and their applications
• Obstruction theory and the study of extensions of maps using relative groups
• Relative versions of other homology and cohomology theories (singular, cellular, de Rham, etc.)

Computational Techniques

• Mayer-Vietoris sequences for relative homology and cohomology
  • Relate relative groups of a union to relative groups of the parts
  • Useful for decomposing spaces and computing relative groups inductively
• Excision and long exact sequences as computational tools
• Relative simplicial and cellular homology for computations in specific cases
  • Relative simplicial homology for simplicial pairs $(X,A)$ where $A$ is a simplicial subcomplex
  • Relative cellular homology for CW pairs $(X,A)$ where $A$ is a subcomplex
• Relative singular homology and cohomology for more general pairs
• Comparison theorems relating different types of relative homology and cohomology
• Use of algebraic tools (exact sequences, spectral sequences, etc.) in computations

Connections to Other Topics

• Relationships between relative homology, cohomology, and other algebraic invariants
  • Relative homotopy groups and their connections to relative homology via the Hurewicz theorem
  • Relative bordism groups and their role in the study of manifolds with boundary
• Poincaré duality and Lefschetz duality as bridges between absolute and relative theories
• Cap and cup products, and their interpretations in terms of relative groups
• Relative versions of other homology and cohomology theories (K-theory, bordism, etc.)
• Applications of relative techniques in other areas of mathematics
  • Algebraic geometry: relative cohomology of algebraic varieties and schemes
  • Differential geometry: relative de Rham cohomology and Hodge theory
  • Topological quantum field theories and their formulation using relative groups
• Connections to categorification and higher categorical structures in topology