Ad
related to: homology theory an introduction to algebraic topology 5th editionchegg.com has been visited by 10K+ users in the past month
Search results
Results from the WOW.Com Content Network
It has played an enormous role in algebraic topology. Its influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations.
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages.The most direct usage of the term is to take the homology of a chain complex, resulting in a sequence of abelian groups called homology groups.
A "homology-like" theory satisfying all of the Eilenberg–Steenrod axioms except the dimension axiom is called an extraordinary homology theory (dually, extraordinary cohomology theory). Important examples of these were found in the 1950s, such as topological K-theory and cobordism theory , which are extraordinary co homology theories, and ...
Peter J. Hilton, An introduction to homotopy theory, Cambridge Tracts in Mathematics and Mathematical Physics, no. 43, Cambridge University Press, 1953. [36] ISBN 0-521-05265-3 MR 0056289; Peter J. Hilton, Shaun Wylie, Homology theory: An introduction to algebraic topology, Cambridge University Press, New York, 1960. [37] ISBN 0-521-09422-4 MR ...
The conclusion of the theorem can equivalently be formulated as: "is an open map".Normally, to check that is a homeomorphism, one would have to verify that both and its inverse function are continuous; the theorem says that if the domain is an open subset of and the image is also in , then continuity of is automatic.
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. Although algebraic topology primarily uses algebra to study topological ...
Homological algebra is a collection of algebraic techniques that originated in the study of algebraic topology but has also found applications to group theory and algebraic geometry The main article for this category is Homological algebra .
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space, the so-called homology groups (). Intuitively, singular homology counts, for each dimension n {\displaystyle n} , the n {\displaystyle n} -dimensional holes of a space.
Ad
related to: homology theory an introduction to algebraic topology 5th editionchegg.com has been visited by 10K+ users in the past month