Complete Algebraic Topology Cheat Sheet: From Fundamentals to Applications

Introduction to Algebraic Topology

Algebraic Topology studies topological spaces by associating algebraic objects (like groups and rings) to them in a way that topological properties are reflected in algebraic properties. This powerful approach allows us to classify spaces, detect holes and obstructions, and solve problems that would be difficult using purely topological methods. Algebraic topology has fundamental applications in mathematics (geometry, analysis), physics (string theory, quantum field theory), computer science (data analysis, networking), and biology (protein structure).

Core Concepts and Foundations

Topological Spaces

• Topological space: A set X with a collection of open sets satisfying specific axioms
• Homeomorphism: A continuous bijection with continuous inverse; preserves topological properties
• Homotopy: A continuous deformation between maps; f ≃ g means f can be continuously deformed into g
• Homotopy equivalence: X ≃ Y means spaces can be continuously deformed into each other

Fundamental Constructions

• Product space: X × Y with the product topology
• Quotient space: X/~ where points are identified according to an equivalence relation
• Mapping cylinder: M(f) = (X × [0,1] ⊔ Y)/(x,1) ~ f(x)
• Mapping cone: C(f) = M(f)/X × {0}
• Suspension: ΣX = (X × [0,1])/(X × {0}, X × {1})

Homotopy Theory

Fundamental Group

• Definition: π₁(X,x₀) = set of based homotopy classes of loops at x₀
• Group operation: Concatenation of loops [f]·[g] = [f·g]
• Functoriality: A continuous map f: X → Y induces a homomorphism f₊: π₁(X,x₀) → π₁(Y,f(x₀))
• Path-connectedness: If X is path-connected, π₁(X,x₀) ≅ π₁(X,x₁) for any x₀, x₁ ∈ X

Higher Homotopy Groups

• Definition: πₙ(X,x₀) = [Sⁿ, X]₊ (based homotopy classes of maps from n-sphere to X)
• Abelian groups: πₙ(X,x₀) is abelian for n ≥ 2
• Long exact sequence: If (X,A) is a pair, there’s a long exact sequence: ···→πₙ(A)→πₙ(X)→πₙ(X,A)→πₙ₋₁(A)→···

Key Results and Techniques

• Seifert-van Kampen Theorem: Computes π₁(X ∪ Y) from π₁(X), π₁(Y), and π₁(X ∩ Y)
• Whitehead’s Theorem: A map f: X → Y between CW complexes is a homotopy equivalence if it induces isomorphisms on all homotopy groups
• Freudenthal Suspension Theorem: Σ: πₙ(X) → πₙ₊₁(ΣX) is an isomorphism when X is (n-1)-connected and n < 2k-1 (where k is the connectivity of X)

Homology Theory

Simplicial Homology

• n-simplex: Convex hull of n+1 affinely independent points
• Chain complex: C₀ ← C₁ ← C₂ ← ··· with boundary operators ∂ₙ: Cₙ → Cₙ₋₁
• Chain groups: Cₙ(K) = free abelian group generated by n-simplices in K
• Boundary operator: ∂([v₀,…,vₙ]) = Σᵢ(-1)ⁱ[v₀,…,v̂ᵢ,…,vₙ] (v̂ᵢ means omit vᵢ)
• Homology groups: Hₙ(K) = ker(∂ₙ)/im(∂ₙ₊₁)

Singular Homology

• Singular n-simplex: A continuous map σ: Δⁿ → X from standard n-simplex to X
• Chain groups: Cₙ(X) = free abelian group generated by singular n-simplices
• Boundary operator: Similar to simplicial homology
• Homology groups: Hₙ(X) = ker(∂ₙ)/im(∂ₙ₊₁)
• Reduced homology: H̃ₙ(X) = Hₙ(X) for n > 0, and H̃₀(X) = kernel of augmentation

Cellular Homology

• Cell complex: Space built by attaching cells of increasing dimension
• Chain complex: C^cell₀ ← C^cell₁ ← C^cell₂ ← ···
• Chain groups: C^cell_n(X) = free abelian group generated by n-cells
• Boundary map: Defined by the degree of attaching maps
• Cellular homology: Computed using cellular chain complex; isomorphic to singular homology

Key Properties of Homology

• Homotopy Invariance: If X ≃ Y, then Hₙ(X) ≅ Hₙ(Y) for all n
• Excision: Hₙ(X,A) ≅ Hₙ(X-U,A-U) under certain conditions
• Mayer-Vietoris Sequence: If X = A ∪ B, there’s a long exact sequence: ···→Hₙ(A∩B)→Hₙ(A)⊕Hₙ(B)→Hₙ(X)→Hₙ₋₁(A∩B)→···
• Universal Coefficient Theorem: Relates homology with different coefficient groups

Cohomology Theory

Singular Cohomology

• Cochain groups: Cⁿ(X) = Hom(Cₙ(X),G) (G is an abelian group)
• Coboundary operator: δ: Cⁿ(X) → Cⁿ⁺¹(X) defined by δ(φ)(σ) = φ(∂σ)
• Cohomology groups: Hⁿ(X) = ker(δⁿ)/im(δⁿ⁻¹)
• Cup product: ⌣: Hᵖ(X) × Hᵍ(X) → Hᵖ⁺ᵍ(X) makes H*(X) a ring

Other Cohomology Theories

• De Rham cohomology: Based on differential forms
• Čech cohomology: Based on open covers
• Sheaf cohomology: Based on local-to-global principles with sheaves

Key Results

• Poincaré Duality: For a closed oriented n-manifold M, Hᵏ(M) ≅ Hₙ₋ₖ(M)
• Künneth Formula: H*(X×Y) ≅ H*(X) ⊗ H*(Y) under certain conditions
• Alexander Duality: H̃ⁿ(S^(n+k) – X) ≅ H̃ₖ₋ₙ₋₁(X) for X ⊂ S^(n+k)

Computational Methods

Computing Fundamental Groups

1. For simple spaces (spheres, tori): Use direct geometric arguments
2. For CW complexes: Analyze 1-skeleton and relations from 2-cells
3. For unions: Apply Seifert-van Kampen theorem
4. For covering spaces: Use relation to the fundamental group of the base space

Computing Homology Groups

1. Choose appropriate method:

   • Simplicial homology (for triangulable spaces)
   • Cellular homology (for CW complexes)
   • Mayer-Vietoris (for spaces decomposable as union of simpler spaces)

2. Steps for cellular homology:

   • Identify the cellular structure
   • Determine the boundary maps (using degrees of attaching maps)
   • Compute kernels and images
   • Calculate Hₙ = ker(∂ₙ)/im(∂ₙ₊₁)

Computing Cohomology Rings

1. Compute the cohomology groups Hⁿ(X)
2. Determine the cup product structure:
   • Use geometric intuition
   • Apply known results (e.g., for projective spaces)
   • Use cellular cohomology and cellular approximation

Spectral Sequences

Basic Definitions

• Spectral sequence: Collection of pages {E_r} with differentials d_r
• Convergence: E_∞ relates to the desired homology/cohomology
• Filtration: Increasing sequence of subspaces/subgroups

Main Types

• Serre spectral sequence: For fibrations F → E → B
• Leray-Serre spectral sequence: Generalizes Serre’s
• Adams spectral sequence: For computing stable homotopy groups
• Atiyah-Hirzebruch spectral sequence: For generalized (co)homology theories

Applications

• Computing homology of complicated spaces
• Proving theoretical results (e.g., about characteristic classes)
• Relating different cohomology theories

Comparison of Important Spaces

Common Challenges and Solutions

Challenge: Non-computability of Homotopy Groups

• Problem: Higher homotopy groups are difficult to compute directly
• Solutions:
  • Use long exact sequences of fibrations
  • Apply Freudenthal suspension theorem and stabilization
  • Use spectral sequences and obstruction theory

Challenge: Working with Complicated Spaces

• Problem: Direct computation is intractable for complex spaces
• Solutions:
  • Decompose into simpler spaces (use Mayer-Vietoris)
  • Use cellular structures when possible
  • Apply spectral sequences
  • Use computer algebra systems for large calculations

Challenge: Distinguishing Spaces

• Problem: Determining whether spaces are homeomorphic
• Solutions:
  • Compare invariants (homology, cohomology rings, homotopy groups)
  • Look for obstructions to equivalence
  • Apply classification theorems when available

Best Practices and Tips

• Geometric Intuition:

  • Draw pictures and visualize simplices, attaching maps
  • Think about deformation retractions to simpler spaces
  • Use concrete examples to guide abstract reasoning

• Computational Strategies:

  • Always start with the simplest possible approach
  • Check results using multiple methods
  • Use the Universal Coefficient Theorem to switch between homology and cohomology

• When Working with CW Complexes:

  • Minimize the number of cells
  • Choose cell structure that reflects symmetries
  • Use cellular approximation for maps

Applications

Intersection Theory and Fixed Point Theorems

• Intersection number: Value in a homology group measuring intersections
• Lefschetz fixed point theorem: Σ (-1)^i Tr(f*: H_i(X) → H_i(X)) = L(f) counts fixed points
• Degree theory: deg(f) determines how many times a map “wraps” one space around another

Characteristic Classes and Bundles

• Vector bundles: Locally trivial fibrationsn with vector space fibers
• Characteristic classes: Cohomology classes measuring the “twistedness” of bundles
  • Stiefel-Whitney classes (for real bundles)
  • Chern classes (for complex bundles)
  • Pontryagin classes (for real bundles, derived from Chern classes)
• Euler class: Top Stiefel-Whitney/Chern class, generalizes Euler characteristic

Obstruction Theory

• Extension problem: When can a map f: A → Y be extended to f̃: X → Y where A ⊂ X?
• Obstructions: Elements in cohomology groups of X that vanish precisely when extensions exist
• Applications: Embedding problems, vector fields on manifolds, sectionsn of bundles

Further Learning Resources

Textbooks

• “Algebraic Topology” by Allen Hatcher (freely available online)
• “Topology and Geometry” by Glen Bredon
• “Differential Forms in Algebraic Topology” by Bott and Tu
• “A Concise Course in Algebraic Topology” by J. P. May

Online Resources

• Allen Hatcher’s webpage: https://pi.math.cornell.edu/~hatcher/
• nLab: https://ncatlab.org/ (category-theoretic perspective)
• MathOverflow and Math StackExchange for specific problems

Software Tools

• GAP (Groups, Algorithms, Programming)
• Sage (open-source mathematical software)
• Macaulay2 (algebraic geometry system with homological algebra capabilities)
• CHomP (Computational Homology Project)

Advanced Topics for Further Study

• Morse Theory
• K-theory
• Cobordism Theory
• Rational Homotopy Theory
• Persistent Homology and Topological Data Analysis