"Analysis Situs" is a seminal
mathematics paper that
Henri Poincaré published in 1895. Poincaré published five supplements to the paper between 1899 and 1904.
These papers provided the first systematic treatment of
topology and revolutionized the subject by using algebraic structures to distinguish between non-
homeomorphic topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
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 invariants that classify topological spaces up to homeomorphism, though usually most classify ...
.
[Dieudonné 1989: 15–35.] Poincaré's papers introduced the concepts of the
fundamental group and
simplicial homology, provided an early formulation of the
Poincaré duality theorem, introduced the
Euler–Poincaré characteristic 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 included in the kernel of ...
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 by H ...
, which was later proven as a theorem. The 1895 paper coined the mathematical term "
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
".
Footnotes
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Mathematics papers
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