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Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. ... ISBN 978-1-4471-2157-2 ...
24 Number Theory: Algebraic Numbers and Functions, Helmut Koch (2000, ISBN 978-0-8218-2054-4) 25 Dirac Operators in Riemannian Geometry , Thomas Friedrich (2000, ISBN 978-0-8218-2055-1 ) 26 An Introduction to Symplectic Geometry , Rolf Berndt (2001, ISBN 978-0-8218-2056-8 )
His research interests lay in the area of algebra, involving abelian groups, modules, homological algebra, and combinatorics. [5] Rotman was the Managing Editor of the Proceedings of the American Mathematical Society in 1972–1973. [4] In 1985 he was the Annual Visiting Lecturer of the South African Mathematical Society. [6]
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 up to homotopy equivalence .
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
Chain (algebraic topology) Betti number; Euler characteristic. Genus; Riemann–Hurwitz formula; Singular homology; Cellular homology; Relative homology; Mayer–Vietoris sequence; Excision theorem; Universal coefficient theorem; Cohomology. List of cohomology theories; Cocycle class; Cup product; Cohomology ring; De Rham cohomology; Čech ...
For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.