"Analysis Situs" is a seminal

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

paper that

Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...

published in 1895. Poincaré published five supplements to the paper between 1899 and 1904. These papers provided the first systematic treatment of

topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

and revolutionized the subject by using algebraic structures to distinguish between non-

homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

s, founding the field of

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

.Dieudonné 1989: 15–35. Poincaré's papers introduced the concepts of the

fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It record ...

and

simplicial homology In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected component (topology), ...

, provided an early formulation of the

Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ...

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

, introduced the

Euler–Poincaré characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space' ...

for

chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...

es, and raised several important conjectures, including the celebrated

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

, which was later proven as a theorem. The 1895 paper coined the mathematical term "

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

Footnotes

References

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