PHD
  1. Algebra  1.1. Group Theory  1.1.1. Basic Properties of Groups  1.1.2. Subgroups  1.1.3. Cosets and Lagrange's Theorem  1.1.4. Normal Subgroups  1.1.5. Quotient Groups  1.1.6. Group Homomorphisms  1.1.7. Sylow Theorems  1.1.8. Free Groups  1.1.9. Group Actions  1.1.10. Simple and Solvable Groups  1.2. Ring Theory  1.2.1. Rings and Ideals  1.2.2. Ring Homomorphisms  1.2.3. Polynomial Rings  1.2.4. Principal Ideal Domains  1.2.5. Unique Factorization Domains  1.2.6. Noetherian Rings  1.2.7. Artinian Rings  1.3. Field Theory  1.3.1. Field Extensions  1.3.2. Splitting Fields  1.3.3. Galois Theory  1.3.4. Algebraic Closures  1.3.5. Finite Fields  1.3.6. Transcendental Extensions  1.4. Module Theory  1.4.1. Modules over Rings  1.4.2. Free Modules  1.4.3. Module Homomorphisms  1.4.4. Simple and Semi-Simple Modules  1.4.5. Projective Modules  1.4.6. Injective Modules  1.5. Linear Algebra  1.5.1. Vector Spaces  1.5.2. Linear Transformations  1.5.3. Eigenvalues and Eigenvectors  1.5.4. Canonical Forms  1.5.5. Inner Product Spaces  1.5.6. Spectral Theorem  1.5.7. Singular Value Decomposition
  2. Analysis  2.1. Real Analysis  2.1.1. Real Numbers and Completeness  2.1.2. Sequences and Series  2.1.3. Continuity  2.1.4. Differentiation  2.1.5. Riemann Integration  2.1.6. Metric Spaces  2.1.7. Lebesgue Measure  2.2. Complex Analysis  2.2.1. Complex Numbers  2.2.2. Analytic Functions  2.2.3. Contour Integration  2.2.4. Cauchy's Theorem  2.2.5. Residue Theorem  2.2.6. Conformal Mappings  2.3. Functional Analysis  2.3.1. Normed Spaces  2.3.2. Banach Spaces  2.3.3. Hilbert Spaces  2.3.4. Linear Operators  2.3.5. Spectral Theory  2.3.6. Compact Operators  2.4. Measure Theory  2.4.1. Sigma Algebras  2.4.2. Measures and Integrals  2.4.3. Lebesgue Integration  2.4.4. Convergence Theorems  2.4.5. Fubini's Theorem  2.5. Harmonic Analysis  2.5.1. Fourier Series  2.5.2. Fourier Transforms  2.5.3. Distribution Theory  2.5.4. Wavelets
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Continuity  3.1.3. Compactness  3.1.4. Connectedness  3.1.5. Metric Topology  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy Theory  3.2.3. Homology Theory  3.2.4. Cohomology  3.2.5. Exact Sequences  3.3. Differential Topology  3.3.1. Smooth Manifolds  3.3.2. Tangent Spaces  3.3.3. Vector Fields  3.3.4. Differential Forms
  4. Geometry  4.1. Differential Geometry  4.1.1. Curves and Surfaces  4.1.2. Riemannian Geometry  4.1.3. Geodesics  4.1.4. Curvature  4.2. Algebraic Geometry  4.2.1. Affine and Projective Varieties  4.2.2. Schemes  4.2.3. Divisors and Intersections  4.2.4. Cohomology in Algebraic Geometry  4.3. Computational Geometry  4.3.1. Convex Hulls  4.3.2. Triangulations  4.3.3. Voronoi Diagrams  4.3.4. Geometric Algorithms
  5. Number Theory  5.1. Elementary Number Theory  5.1.1. Divisibility  5.1.2. Congruences  5.1.3. Modular Arithmetic  5.2. Analytic Number Theory  5.2.1. Prime Number Theorem  5.2.2. Dirichlet Series  5.2.3. Zeta and L-functions  5.3. Algebraic Number Theory  5.3.1. Number Fields  5.3.2. Ideals  5.3.3. Class Groups  5.3.4. Galois Representations
  6. Combinatorics  6.1. Enumerative Combinatorics  6.1.1. Generating Functions  6.1.2. Recurrence Relations  6.1.3. Permutations and Combinations  6.2. Graph Theory  6.2.1. Graph Isomorphisms  6.2.2. Planarity  6.2.3. Graph Coloring  6.3. Extremal Combinatorics  6.3.1. Ramsey Theory  6.3.2. Probabilistic Methods in Extremal Combinatorics  6.4. Algebraic Combinatorics  6.4.1. Matrix Methods  6.4.2. Representation Theory
  7. Logic and Foundations  7.1. Set Theory  7.1.1. Zermelo-Fraenkel Axioms  7.1.2. Ordinals and Cardinals  7.2. Model Theory  7.2.1. Structures and Theories in Model Theory  7.2.2. Compactness Theorem  7.3. Proof Theory  7.3.1. Formal Systems  7.3.2. Consistency and Completeness
  8. Probability and Statistics  8.1. Probability Theory  8.1.1. Random Variables  8.1.2. Expectation and Variance  8.1.3. Central Limit Theorem  8.2. Stochastic Processes  8.2.1. Markov Chains  8.2.2. Brownian Motion  8.3. Statistical Inference  8.3.1. Hypothesis Testing  8.3.2. Regression Analysis
  9. Applied Mathematics  9.1. Numerical Analysis  9.1.1. Iterative Methods  9.1.2. Interpolation  9.1.3. Error Analysis  9.2. Partial Differential Equations  9.2.1. Elliptic Equations  9.2.2. Parabolic Equations  9.2.3. Hyperbolic Equations  9.3. Dynamical Systems  9.3.1. Chaos Theory  9.3.2. Stability Analysis in Dynamical Systems

PHD → Topology → Algebraic Topology ↓


Exact Sequences


In algebraic topology and other areas of mathematics, exact sequences are a fundamental concept that helps us understand the relationship between different algebraic structures, typically between modules over a ring or groups. This is important for studying the properties of topological spaces, and therefore, exact sequences have a wide range of applications.

Understanding the building blocks

Before diving into exact sequences, it is important to understand the concept of homomorphism and the key concepts of kernel and image. Let's start with these concepts.

Homomorphisms

A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or modules. For example, if f: A to B is a homomorphism between two groups, then it must satisfy the following property:

f(x * y) = f(x) * f(y)

where * denotes the group operation. For rings, homomorphisms must respect addition and multiplication.

Kernel and image

The kernel of a homomorphism f: A to B, denoted ker(f), is the set of elements in A that map to the identity element in B:

ker(f) = { a in A | f(a) = e_B }

where e_B is the identity element in B

The image of a homomorphism, im(f), is the set of elements in B that are mapped to elements in A:

im(f) = { b in B | b = f(a) for some a in A }

What is the exact sequence?

An exact sequence is a sequence of algebraic objects and isomorphisms between them, in which the image of one isomorphism is equal to the kernel of the next isomorphism. In simple terms, the output of one function on the sequence can perfectly serve as the input for the next. The sequence is usually represented as:

... → A_n-1 → A_n → A_n+1 → ...

The sequence is called exact if at each position, the image of the earlier map is the same as the exact kernel of the next:

im(φ_{n-1}) = ker(φ_n)

where φ denotes symmetries.

Types of exact sequences

Short exact sequence

A short exact sequence looks like this:

0 → A → B → C → 0

Here, the sequence begins and ends with the zero object (such as the zero vector space or the trivial group). For this sequence to be exact, the map A → B must have kernel 0 (i.e., be injective), and the map B → C must map onto C (i.e., be surjective).

Long exact sequence

Long exact sequences appear in contexts such as homology and cohomology. For example, given a short exact sequence of chain complexes, there is an induced long exact sequence in homology:

... → H_n(A) → H_n(B) → H_n(C) → H_{n-1}(A) → ...

Visual example

0 F A Yes B H C 0

This SVG representation shows a short exact sequence. The sequence starts at 0, mapping to the collection A, then to B, finally descending to C, and ending back at 0.

Applications in algebraic topology

Exact sequences play an important role in algebraic topology, especially in the study of homology and cohomology theories. Let's explore some typical applications.

Mayer–Vietoris sequence

This long exact sequence is a powerful tool for computing the isomorphisms of a space that can be decomposed into two overlapping subspaces. Given a topological space X and its subspaces U and V such that X = U ∪ V, the Mayer–Vietoris sequence is:

... → H_n(U ∩ V) → H_n(U) ⊕ H_n(V) → H_n(X) → H_{n-1}(U ∩ V) → ...

The Five Lemma and the Snake Lemma

The Five Lemma and the Snake Lemma are important results concerning exact sequences, frequently used in algebraic topology and homological algebra.

The five lemmas help to prove the isomorphism of groups in the middle of a commutative diagram with exact rows, such as:

A → B → C → D → E
↓   ↓   ↓   ↓   ↓
A' → B' → C' → D' → E'

If the first four vertical maps are isomorphisms, the fifth is also an isomorphism.

The snake lemma provides a long exact sequence with exact rows from a commutative diagram:

0 → A → B → C → 0
0 → A' → B' → C' → 0

This proves the existence of an induced long exact sequence connecting the kernel and cokernel of the corresponding maps.

Further consideration

Understanding exact sequences opens the door to higher abstraction levels in algebraic topology and homological algebra. They allow us to study complex algebraic structures and obtain various fascinating results.

Learning to manipulate and reason with these sequences is an important skill for a mathematician working in algebraic topology, since they provide insight into the "holes" and combinatorial properties of places in more concrete algebraic terms.

The connections made through exact sequences underscore the beauty and elegance of mathematical structures. They demonstrate how topological, algebraic, and geometric concepts intertwine to form the rich fabric of mathematical theory.

Conclusion

Exact sequences are a powerful analytical tool in mathematics, providing a unified language for discussing the properties of algebraic structures, especially in algebraic topology. Through various theories and examples, their applicability extends beyond the theoretical to practical scenarios, enriching our understanding of mathematical and topological constructions.


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