maths.free › Algebraic Topology

Algebraic Topology

Fundamental groups, homology: counting holes with algebra.

Opetuksia

Advanced The fundamental group Loops up to deformation; the circle's winding number; van Kampen. polar form of -1 Advanced Homology and the Euler characteristic Counting holes in every dimension with chain complexes. 6 - 12 + 8

Attach a group to a space so that continuous maps become homomorphisms. Then the question "are these two spaces the same shape?" becomes an algebra computation, which is how we know a doughnut is not a sphere.

Lisää oppitunteja

Homotopy

Algebraic topology Homotopy Covering space Seifert–Van Kampen theorem Homotopy group Simply connected space Deformation retract

Homology

Homology (mathematics)Simplicial complex Singular homology Chain complex Exact sequence Betti number Euler characteristic CW complex Mayer–Vietoris sequence Cohomology Poincaré duality

Consequences

Brouwer fixed-point theorem Borsuk–Ulam theorem Hairy ball theorem Jordan curve theorem Poincaré conjecture Classification of manifolds

Osa näistä opetuksista on mukautettu avoimiin teksteihin: Wikipedia. Jokainen oppituntisivu nimeää sen lähteen ja lisenssin.

Symbolit, joita käytetään täällä

Napauta symbolia koko määrittelyn, kuvan ja sen jokaisen kirjaimen osalta.

Kysymyksiä

What does the fundamental group measure?

Loops in a space up to deformation. In a plane every loop shrinks to a point (trivial group); around a hole a loop can wind n times (the integers).

Onko sinulla oma ongelma?
Kirjoita se tai piirrä se ratkaisijan sivulle: jokainen vaihe, kuva, vastaus piilotettuna, kunnes kysyt. Ratkaise →

Muut sivukonttorit

＋ Arithmetic 𝑥 Algebra △ Geometry ∿ Trigonometry ƒ Precalculus σ Statistics & Probability ⊢ Discrete Math & Logic ∫ Calculus ⊞ Linear Algebra ℕ Number Theory ẏ Differential Equations ∇ Multivariable Calculus ε Real Analysis ⋔ Combinatorics & Graph Theory ∈ Set Theory & Logic 𝔾 Abstract Algebra 𝑃 Probability Theory ≈ Numerical Methods ℂ Complex Analysis ◎ Topology μ Measure Theory ‖·‖ Functional Analysis κ Differential Geometry → Category Theory ∂ Partial Differential Equations ≋ Fluid Dynamics ∞ Frontiers