Njia za utafitiGeometry & TopologySomo linapatikana

Stable homotopy theory and generalised cohomology

Optional research direction: what survives suspension. Spectra and the stable homotopy category, Brown representability, generalised cohomology theories (K-theory and cobordism), Thom's computation of cobordism, the Adams spectral sequence and a glimpse of chromatic homotopy theory.

math.geom.stable-homotopy-theory

Fungua somo Onyesha kila kinachohitajika kufikia mada hii

Inajengwa juu ya

Homotopy theory Vector bundles and characteristic classes

Masharti ya awali ya lazima 51 (hadi chini kabisa)

Utajifunza

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.

Fasili

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.

Nadharia

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.

Mawazo potofu ya kawaida

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.

Ugunduzi wa hiari

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

Usomaji zaidi

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