maths.free › Algebraic Topology

Algebraic Topology

Fundamental groups, homology: counting holes with algebra.

Lekcije

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.

Više lekcija

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

Neke od tih lekcija prilagođene su na otvorenim tekstovima: Wikipedia. Svaka lekcija naziva svoj izvor i dozvolu.

Simboli koji se koriste ovdje

Dodirnite bilo koji simbol za punu definiciju, sliku, i što znači svako slovo u njoj.

Pitanja koja ljudi postavljaju

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).

Imaš li problema?
Upišite ili nacrtajte na stranici rješavača: svaki korak, sliku, odgovor skriven dok ne pitate. Riješi →

Ostale grane

＋ 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