SYSTEMATIC MATHEMATICS

Stable Homotopy Theory & Generalized Cohomology

Use 127 visible bilingual chapters in thirteen content-sized units to build stable homotopy theory from based spaces, suspension, loops, fibers, cofibers, sequential spectra, model discipline, the stable category, finite duality, stable stems, generalized homology and cohomology, Brown representability, Steenrod operations, ring and module spectra, K-theory, cobordism, exact couples, AHSS and Adams spectral sequences, chromatic height, Morava theories, nilpotence, periodicity, Bousfield localization, completion, fracture squares, and carefully bounded equivariant, parametrized, motivic, stable-infinity, and spectral-algebra interfaces. Every symbol receives a local meaning; every chapter includes an exact definition, theorem or bounded claim, a complete proof route whose length follows the argument, a failure boundary, a worked calculation, a diagnostic question, matched practice, and a solution. Ten original reproducible dossiers retain typed arrows, finite data, degree conventions, exact arithmetic, unknown exponents, and provenance. This is a Stage 5 doctoral specialization core and research preparation—not complete model-category foundations, deep chromatic computation, full equivariant or motivic theory, telescope-comparison results, tmf, surgery, Floer theory, or universal research-frontier mastery.

Before this course: Completed Algebraic Topology; Category Theory, Homological Algebra & Derived Methods; Abstract Algebra; Linear Algebra; Proof, Logic & Set Theory; Topology; and familiarity with vector bundles, characteristic classes, exact sequences, chain complexes, tensor products, and spectral-sequence notation.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Pointed spaces and maps that preserve the basepoint

Objective: Why does specifying f(+1) determine a based map S⁰→X, while it does not determine an unbased map?

A pointed space is not merely a space with a decorative dot. It is a pair (X,x₀), and x₀ is part of the data. A based map f:(X,x₀)→(Y,y₀) must satisfy f(x₀)=y₀. This one equation determines which constant map is allowed, which homotopies count, and where later loop and smash constructions attach.

Write Map_*(X,Y) for based maps and [X,Y]_* for based homotopy classes. The star is a condition, not multiplication. A based homotopy H:X×I→Y keeps H(x₀,t)=y₀ for every t. An ordinary homotopy may move the image of x₀ and therefore need not prove equality in [X,Y]_*.

The zero object in pointed spaces is the one-point space *. There is exactly one based map *→X and exactly one based map X→*. This makes kernels, cones, wedges, and reduced constructions behave more like algebraic objects than their unbased counterparts, but it does not turn the category of pointed spaces into an abelian category.

The based zero-sphere S⁰ has two points {−1,+1} and basepoint −1. A based map S⁰→X is completely determined by the image of +1 because −1 is forced to x₀. Thus Map_*(S⁰,X) is naturally identified with the underlying set of points of X. This small model will later explain why S⁰ acts as the unit for smash product.

Whenever a formula uses a star, record three facts first: which object carries the basepoint, which map must preserve it, and whether the homotopy also preserves it at every time. Stable homotopy theory suppresses these facts typographically because they occur constantly; this course keeps them explicit until the type check becomes automatic.

For every pointed space (X,x₀), evaluation at the nonbasepoint +1 gives a natural bijection Map_*(S⁰,X)≅X. Under this bijection the constant based map corresponds to x₀.

Define ev₊(f)=f(+1). This is well typed because +1 is a point of S⁰ and f lands in X.

For x∈X, define f_x(−1)=x₀ and f_x(+1)=x. The domain is discrete, so f_x is continuous, and it is based by construction.

The composites satisfy ev₊(f_x)=x and f_{ev₊(f)}=f because a based map already has f(−1)=x₀. Naturality follows from h∘f_x=f_{h(x)} for every based h:X→Y.

Let X={x₀,a,b} be a discrete pointed space with basepoint x₀. List every based map S⁰→X and identify the constant based map.

  1. The basepoint condition forces f(−1)=x₀ in every case.
  2. The remaining point +1 may map independently to x₀, a, or b.
  3. Thus the maps are f_{x₀}, f_a, and f_b. Only f_{x₀} sends both domain points to x₀.

Result: There are exactly three based maps, corresponding to x₀,a,b under evaluation at +1; f_{x₀} is the constant based map.