The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology.
For a non-intersecting graph in the Euclidean plane, with V vertices (or corners), E edges and F faces (counting the exterior) Euler showed that V-E+F= 2.
The Euler characteristic of other surfaces is a useful topological invariant, which has been extended to higher dimensions using Betti numbers.
The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials.
As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure.
Johann Benedict Listing, inventor of the word "topology", wrote an 1847 paper "Vorstudien zur Topologie" in which he defined a "complex".
After Abel, Jacobi, and Riemann, some of the most important contributors to the theory of abelian functions were Weierstrass, Frobenius, Poincaré and Picard.
Henri Poincaré's 1895 paper Analysis Situs studied three-and-higher-dimensional manifolds (which he called "varieties"), giving rigorous definitions of homology, homotopy, and Betti numbers and raised a question, today known as the Poincaré conjecture, based his new concept of the fundamental group.
In 2003, Grigori Perelman proved the conjecture using Richard S. Hamilton's Ricci flow, this is after nearly a century of effort by many mathematicians.
During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through differential geometry and Lie group theory.
Eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex tori.