maths.free › Algebraic Topology

Algebraic Topology

Fundamental groups, homology: counting holes with algebra.

Àwọn Àkọsílẹ̀

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.

Àwọn ìwé-ìwé mìíràn

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

Awọn ẹkọ diẹ ninu awọn wọnyi ni a ṣe atunṣe lati awọn ọrọ ti o ṣii: Wikipedia. Àwọn ojú-ìwé ìwé-ìwé fún àwọn àwọn ìṣàmúlò-ètò àti àwọn ìṣàmúlò-ètò.

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Àwọn Àtòjọ-ẹ̀yàn

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).

Ní àwọn ààyè rẹ̀?
Ṣàfihàn àti àwòrán nínú ojú-ìwé ìṣàmúlò-ètò: àwọn ìsẹ̀ gbogbo, àwòrán, àwọn à Àwọn Àwọn Àkọ́lé →

Àwọn ìṣàmúlò-ètò mìíràn

＋ 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