Stable homotopy theory and generalised cohomology

前置知识

51 项必需的前置知识（一直追溯到底）

你将学会

• Explain stabilisation through the Freudenthal suspension theorem and define spectra informally and formally.
• State Brown representability and give K-theory and cobordism as examples of generalised cohomology theories.
• Describe Thom's identification of cobordism with homotopy groups of Thom spectra and the role of the Adams spectral sequence.

定义

• A spectrum is a sequence of pointed spaces Eₙ with maps ΣEₙ → Eₙ₊₁.
• A generalised cohomology theory satisfies the Eilenberg–Steenrod axioms except the dimension axiom.
• Two closed n-manifolds are cobordant if together they form the boundary of a compact (n + 1)-manifold.

定理

• Brown representability: every generalised cohomology theory on CW complexes is represented by a spectrum.
• Thom: the unoriented cobordism ring is isomorphic to the homotopy of the Thom spectrum MO, a polynomial ring over 𝔽₂.
• Bott periodicity: πₙ₊₂(BU × ℤ) ≅ πₙ(BU × ℤ), making complex K-theory 2-periodic.
• The stable homotopy groups of spheres are finite in positive degrees (Serre), yet are known only in a limited range.

常见误解

• Thinking the stable homotopy groups of spheres are known; they are computed only in a range, and their structure is the subject of ongoing research.
• Believing generalised cohomology theories are just homology with coefficients; K-theory and cobordism are not ordinary cohomology with any coefficients.

可选发现

• Stable ∞-categories The stable homotopy category is the homotopy category of the stable ∞-category of spectra, and it is triangulated.
• Arithmetic geometry and elliptic curves Chromatic homotopy theory is organised by formal groups, objects first studied in number theory and arithmetic geometry.
• ← Vector bundles and characteristic classes

延伸阅读

• J. F. Adams, Stable Homotopy and Generalised Homology (1974)
• Douglas C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres (1986)