Examples

This gallery walks through every public feature of mathematicskit.topology: simplicial complexes and their homology, the fundamental group, the classification of surfaces, fixed-point theorems, invariants of curves, and the topological analysis of point clouds.

Each script in this gallery is self-contained and can be run directly with python examples/topology/<section>/<script>.py.

Sections

• curves – Gauss’s linking number and Hopf’s turning number.

• surfaces – the Möbius strip, the Klein bottle, the classification of surfaces, and critical points of height functions.

• homology – the Smith normal form, Betti numbers, Poincaré’s homology and the Euler-Poincaré formula, and the fundamental group.

• complexes – barycentric subdivision, products and the Künneth formula, and the Mayer-Vietoris sequence.

• fixed_points – the Poincaré-Hopf index, Brouwer’s fixed-point theorem, the Lefschetz number, and Sperner’s lemma.

• data – Vietoris-Rips and Čech complexes, persistent homology, its stability, and Mapper.

Complexes

Constructions on simplicial complexes: subdivision, products, and gluing.

Brouwer, Alexander, and the invariance of homology under subdivision (1911-1915)

Brouwer, Alexander, and the invariance of homology under subdivision (1911-1915)

The Künneth formula for products (1923)

The Künneth formula for products (1923)

The Mayer-Vietoris sequence (1929-1930)

The Mayer-Vietoris sequence (1929-1930)

Curves

Topological invariants of closed curves: linking and turning numbers.

Gauss’s linking number (1833)

Gauss's linking number (1833)

Hopf’s Umlaufsatz: the turning number of a closed curve (1935)

Hopf's Umlaufsatz: the turning number of a closed curve (1935)

Topological data analysis

Complexes built from point clouds, persistent homology, and Mapper.

Vietoris’s complex of a point cloud (1927)

Vietoris's complex of a point cloud (1927)

Čech complexes and the nerve theorem (1932-1948)

Čech complexes and the nerve theorem (1932-1948)

Persistent homology (Edelsbrunner, Letscher and Zomorodian 2002)

Persistent homology (Edelsbrunner, Letscher and Zomorodian 2002)

Stability of persistence diagrams (Cohen-Steiner, Edelsbrunner and Harer 2007)

Stability of persistence diagrams (Cohen-Steiner, Edelsbrunner and Harer 2007)

Mapper (Singh, Mémoli and Carlsson 2007)

Mapper (Singh, Mémoli and Carlsson 2007)

Fixed points

Vector-field indices, Brouwer, Lefschetz, and Sperner.

The Poincaré-Hopf index theorem (1885-1927)

The Poincaré-Hopf index theorem (1885-1927)

Brouwer’s fixed-point theorem (1911)

Brouwer's fixed-point theorem (1911)

Lefschetz’s fixed-point theorem and Hopf’s trace formula (1926-1929)

Lefschetz's fixed-point theorem and Hopf's trace formula (1926-1929)

Sperner’s lemma (1928)

Sperner's lemma (1928)

Homology

The Smith normal form, Betti numbers, homology with torsion, and the fundamental group.

Smith’s normal form of an integer matrix (1861)

Smith's normal form of an integer matrix (1861)

Betti’s connectivity numbers (1871)

Betti's connectivity numbers (1871)

Poincaré’s Analysis Situs: homology and the Euler-Poincaré formula (1895-1900)

Poincaré's Analysis Situs: homology and the Euler-Poincaré formula (1895-1900)

Poincaré’s fundamental group (1895)

Poincaré's fundamental group (1895)

Surfaces

One-sided surfaces, the Klein bottle, the classification theorem, and critical points.

Möbius and Listing: the one-sided surface (1858-1865)

Möbius and Listing: the one-sided surface (1858-1865)

Klein’s bottle and torsion in homology (1882)

Klein's bottle and torsion in homology (1882)

The classification of surfaces (Dehn and Heegaard 1907, Brahana 1921)

The classification of surfaces (Dehn and Heegaard 1907, Brahana 1921)

Morse theory and Banchoff’s critical points (1925-1967)

Morse theory and Banchoff's critical points (1925-1967)