Cobordism and Characteristic Numbers

Цели урока

• Understand the cobordism relation and the ring structure of Omega_*
• Compute Stiefel-Whitney and Pontryagin numbers as cobordism invariants
• Apply the Hirzebruch signature theorem to 4-manifolds
• Understand the Pontryagin-Thom construction linking cobordism to homotopy

Предварительные знания

• Characteristic classes of fiber bundles
• Homology and cohomology
• Manifolds and orientability

• Characteristic Classes
• Homology Groups

When are two manifolds the same from the perspective of bounding a higher-dimensional manifold? Cobordism answers this - and the signature of 4-manifolds distinguishes exotic smooth structures on R^4.

• Quantum gravity: cobordism of 4-manifolds describes transitions between spacetime configurations
• TQFT: topological quantum field theories are functors from the cobordism category to vector spaces
• 4-manifold topology: Donaldson's theorem uses signature to discover exotic smooth structures on R^4
• Condensed matter: cobordism ring classifies topological phases of matter with given symmetry

From Pontryagin to Thom

Lev Pontryagin in 1938 connected homotopy groups of spheres to differential topology. Rene Thom in 1954 systematized cobordism theory and received the Fields Medal for computing the cobordism rings. Friedrich Hirzebruch in 1953 proved the signature theorem via L-classes. Atiyah and Singer in 1963 generalized this to all elliptic operators, unifying these results in the index theorem.

Cobordism and Topological Invariants of Manifolds

The Feynman program in quantum gravity (IAS, 2020) uses cobordism classification of 4-manifolds: two manifolds are cobordant if and only if all their Pontryagin numbers agree - an obstruction to a unified field theory. The same cobordism ring classifies topological phases in condensed matter physics.

Hirzebruch Signature Theorem

The signature sigma(M) of a 4-manifold M is the signature of the intersection form on H^2(M; R). Donaldson's theorem (1983) classified simply connected 4-manifolds by their intersection forms - the signature was the key invariant. Exotic smooth structures on R^4 were discovered using signature arguments.

Pontryagin-Thom Construction

Rene Thom's 1954 construction reduces the computation of cobordism groups to the computation of homotopy groups of spheres - a problem algebraic topologists already know how to attack. This bridge between smooth geometry and homotopy theory is one of the deepest structural results in topology.

First few cobordism groups: Omega_0 = Z (count points mod 2... actually = Z over oriented), Omega_1 = 0, Omega_2 = 0, Omega_3 = 0, Omega_4 = Z (generated by CP^2). The first non-trivial oriented cobordism group is Omega_4, where sigma is the complete invariant.

Connections to other topics

Cobordism unites differential topology, algebraic topology, and mathematical physics.

• TQFT — Related topic
• 4-manifolds — Related topic
• Homotopy spectra — Related topic
• Atiyah-Singer theorem — Related topic

Итоги

• Cobordism: M_1 ~ M_2 iff exists W with delta W = M_1 disjoint union M_2; classes form ring Omega_*
• Stiefel-Whitney numbers w_I[M] are complete Z/2 cobordism invariants (Pontryagin-Thom)
• Signature sigma(M^{4k}) is an oriented cobordism invariant; sigma(M # N) = sigma(M) + sigma(N)
• Hirzebruch theorem: sigma(M^{4k}) = integral over M of L(TM)
• Pontryagin numbers p_I[M] are complete rational oriented cobordism invariants
• Thom isomorphism: Omega_n^O = pi_{n+N}(MO(N)) - cobordism as stable homotopy

Вопросы для размышления

• Why do Stiefel-Whitney numbers live in Z/2 while Pontryagin numbers live in Z? What does this say about orientability?
• How does the Pontryagin-Thom construction translate geometric cobordism into an algebraic homotopy problem?
• Why is sigma additive for oriented cobordism but fails to be a complete invariant over Z (unlike over Q)?

Связанные уроки

• top-24-chern — Characteristic classes build the characteristic numbers of cobordism
• top-26-k-theory — K-theory is the next step after cobordism in generalized cohomology
• top-23 — Homology and fundamental class needed for Stiefel-Whitney numbers
• aa-20-homological