homotopy

History

The concept was introduced by Henri Poincaré in the late 19th century in his works on qualitative geometry. He applied it to classify surfaces.

How it works

Two loops on a surface are homotopic if one can be shrunk to the other without leaving the surface. Analogy: like a rubber band on a globe: can it be shrunk to a point without slipping off the sphere.

History

The concept was introduced by Henri Poincaré in the late 19th century in his works on qualitative geometry. He applied it to classify surfaces.

How it works

Two loops on a surface are homotopic if one can be shrunk to the other without leaving the surface. Analogy: like a rubber band on a globe: can it be shrunk to a point without slipping off the sphere.

History

The origins of the concept go back to Henri Poincaré's 1895 work on analysis situs, where he defined the fundamental group. Further development is associated with Emmy Noether, who contributed to the algebraization of topology, and L.E.J. Brouwer, who refined the concept. By the 1930s, Witold Hurewicz defined higher homotopy groups. In the second half of the 20th century, homotopy type theory, proposed by Voevodsky, unified homotopy with logic and computer science, gaining recognition in the 2010s.

How it works

The fundamental group π₁(X) consists of homotopy classes of loops with composition operation. Higher homotopy groups π_n(X) generalize to spheres S^n. Homotopy groups are computed using exact sequences, spectral sequences, and the Hurewicz theorem. For example, π₃(S²)=ℤ is generated by the Hopf map. Homotopy invariance is connected to categorical properties; Quillen's model category theory formalizes homotopy structures. The principle of superposition and continuity underlies deformations, and Noether's theorem on symmetries is reflected in invariants.

Open problems & fun facts

Computation of homotopy groups of spheres in general; Blake–Sutherland conjecture on homotopy groups of products; Resolution of problems in homotopy type theory, such as the univalence axiom

💡 Homotopy groups of spheres are one of the greatest unsolved problems in topology. For example, mathematicians still do not know all homotopy groups of spheres of dimension higher than 2.