Henri PoincaréJules Henri Poincaré (UKˈpwæ̃kɑreɪ, ; ɑ̃ʁi pwɛ̃kaʁe; 29 April 1854 - 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics.
Singular homologyIn algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups Intuitively, singular homology counts, for each dimension n, the n-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions (see also the related theory simplicial homology).
Betti numberIn algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of n-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicial complexes or CW complexes), the sequence of Betti numbers is 0 from some point onward (Betti numbers vanish above the dimension of a space), and they are all finite.
Exact sequenceAn exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an ) such that the of one morphism equals the kernel of the next. In the context of group theory, a sequence of groups and group homomorphisms is said to be exact at if . The sequence is called exact if it is exact at each for all , i.e., if the image of each homomorphism is equal to the kernel of the next. The sequence of groups and homomorphisms may be either finite or infinite.
Möbius stripIn mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from the third century CE. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns.
Free abelian groupIn mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis.
Fundamental groupIn the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups.
TorusIn geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus.
Finitely generated abelian groupIn abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set of or that generate . Every finite abelian group is finitely generated. The finitely generated abelian groups can be completely classified. The integers, , are a finitely generated abelian group. The integers modulo , , are a finite (hence finitely generated) abelian group.
CW complexA CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The C stands for "closure-finite", and the W for "weak" topology.
