topology

History

The birth of topology is associated with Henri Poincaré, who in 1895 published 'Analysis situs', laying the foundations of algebraic topology. He also formulated the famous Poincaré conjecture about the three-dimensional sphere, proved only in 2003.

How it works

Topology works through invariants — like passport data of a shape that do not change under stretching. For example, the number of holes (birational genus) or the Euler characteristic: for a cube it is 2 (V-E+F=8-12+6=2), same as for a sphere, while for a torus it is 0. If invariants differ, the objects are definitely not equivalent.

History

The birth of topology is associated with Henri Poincaré, who in 1895 published 'Analysis situs', laying the foundations of algebraic topology. He also formulated the famous Poincaré conjecture about the three-dimensional sphere, proved only in 2003.

How it works

Topology works through invariants — like passport data of a shape that do not change under stretching. For example, the number of holes (birational genus) or the Euler characteristic: for a cube it is 2 (V-E+F=8-12+6=2), same as for a sphere, while for a torus it is 0. If invariants differ, the objects are definitely not equivalent.

History

The roots of topology go back to Euler's solution of the Königsberg bridge problem (1736) and his polyhedron theorem (1758). Henri Poincaré (included in the scientists list) laid the foundations in a series of works 'Analysis situs' (1895–1904), introducing the fundamental group and homologies. In the 20th century, the theory flourished thanks to Brouwer, Hurewicz, Whitehead, and others; the Poincaré conjecture was proved by Perelman in 2003.

How it works

Topological invariants extract algebraic information from a space. For example, the fundamental group π₁(X) encodes all possible loops up to continuous deformation: for a sphere, π₁ = {0}; for a circle, π₁ = ℤ. Homologies Hₙ(X) count 'holes' of various dimensions. For manifolds, the de Rham theorem, linking differential forms and cohomology, is often used. In physics, topology manifests through quantum anomalies, solitons, and string theory: Calabi-Yau spaces have non-trivial topology affecting the compactification of extra dimensions.

Open problems & fun facts

Classification of smooth structures on four-dimensional manifolds (the exotic R⁴ problem); The Hodge conjecture on the relation between algebraic cycles and cohomologies (one of the Millennium Problems); Computing homotopy groups of higher-dimensional spheres in general

💡 Topologists have proved that you can turn a sphere inside out without tearing, simply by passing the surface through itself — this is called Smale's sphere eversion.