Analysis Situs is an influential mathematical paper (and a series of addendums) written by Henri Poincare. Besides providing the first systematic treatment of topology, Poincaré revolutionized the subject by using algebraic structures to distinguish between non-homeomorphic topological spaces, effectively creating the subject of algebraic topology. Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. ... Henri Poincaré, photograph from the frontispiece of the 1913 edition of Last Thoughts Jules Henri Poincaré (April 29, 1854 – July 17, 1912) was one of Frances greatest mathematicians, theoretical scientists and a philosopher of science. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... This word should not be confused with homomorphism. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

In the papers he introduced the concepts of fundamental group and simplicial homology. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homeos = identical and logos = word) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

References

• Henri Poincaré, Analysis Situs, Journal de l'Ecole Polytechnique ser 2, 1 (1895) pages 1-123.
• Henri Poincaré, Complément à l'Analysis Situs, Rendiconti del Circolo matematico di Palermo, 13 (1899) pages 285-343.
• Henri Poincaré, Second complément à l'Analysis Situs, Proceedings of the London Mathematical Society, 32 (1900), pages 277-308.
• Henri Poincaré, Sur certaines surfaces algébriques ; troisième complément à l'Analysis Situs, Bulletin de la Société mathématique de France, 30 (1902), pages 49-70.
• Henri Poincaré, Sur les cycles des surfaces algébriques ; quatrième complément à l'Analysis Situs, Journal de mathématiques pures et appliquées, 5° série, 8 (1902), pages 169-214.
• Henri Poincaré, Cinquième complément à l'analysis situs, Rendiconti del Circolo matematico di Palermo 18 (1904) pages 45-110.