maths.free › Algebraic Topology

Algebraic Topology

Fundamental groups, homology: counting holes with algebra.

Kennslustund

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.

Fleiri kennslustundir

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

Sumir af þessum kennslustundum eru aðlagaðar frá opnum textum: Wikipedia. Hver kennslustundarsíða nefnir uppruna sinn og leyfi.

Tákn notuð hér

Pikkaðu á hvaða tákn sem er fyrir fulla skilgreiningu, mynd og hvað hvert bréf í því merkir.

Spurningar sem fólk spyr

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).

Hafa vandamál af eigin þinn?
Sláðu það inn eða teikna það á lesara síðu: hvert skref, mynd, svarið falið þar til þú spyrð. Leysa →

Aðrar útibú

＋ 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