Algebraic Topology: Homology and Cohomology

Henri Poincare introduced Betti numbers in 1895, founding algebraic topology. For a century, those numbers remained the primary tool for classifying spaces. Homology converted topology from geometry into algebra.

• Manifold classification: Betti numbers and the Hirzebruch signature are surgery invariants
• Data science: persistent homology analyzes the shape of point clouds in TDA
• Physics: homology of coherent sheaves is the language of mirror symmetry and string theory

Предварительные знания

• Previous lesson

Simplicial and Singular Homology

Henri Poincare in 1895, in 'Analysis Situs', introduced Betti numbers β_n as the ranks of H_n(X;Q) -- and thereby founded algebraic topology. For the torus T², the Betti numbers are β_0=1, β_1=2, β_2=1, giving χ(T²)=0. Simplicial homology: given a simplicial complex K, form the chain complex C_n(K) with differential ∂_n: C_n → C_{n-1}, ∂_{n-1}∘∂_n = 0. Then H_n(K) = ker(∂_n)/im(∂_{n+1}).

The Cohomology Ring and Poincare Duality

Rene Thom in 1954 introduced the cobordism ring and showed that characteristic classes (Stiefel-Whitney classes) are completely determined by the ring structure of H*(BO;Z/2) ≅ Z/2[w_1,w_2,...]. The cup product ⌣: H^p(X) ⊗ H^q(X) → H^{p+q}(X) makes H*(X;R) a graded commutative ring: α ⌣ β = (-1)^{pq} β ⌣ α.

Key Ideas

• H_n(X) = ker d_n / im d_{n+1} -- a topological invariant of the space
• H*(S^n): Z in degrees 0 and n, zero elsewhere
• Kunneth formula: H_n(X x Y) = sum_{p+q=n} H_p(X) tensor H_q(Y)
• Cohomology ring H*(X;R) with cup product -- a contravariant invariant
• Poincare duality: H^k(M) isomorphic H_{n-k}(M) for closed oriented n-manifolds

Further Directions

The concepts studied here open paths to deeper areas of mathematics.

• aa-29-elliptic-curves — extends

Вопросы для размышления

• Work through a concrete computation using the methods from this lesson.
• How do the ideas here connect to other areas of mathematics covered earlier?