Homology (Topology)
Historically, the term ``homology'' was first used in a topological sense by Poincaré. 
To him,it meant pretty much what is now called a Cobordism, meaning that a homology was thought of as a relation betweenManifolds mapped into a Manifold. Such Manifolds form a homology when theyform the boundary of a higher-dimensional Manifold inside the Manifold in question.
To simplify thedefinition of homology, Poincaré simplified the spaces he dealt with. He assumed that all the spaces he dealt with had atriangulation (i.e., they were ``Simplicial Complexes''). Then instead of talking aboutgeneral ``objects'' in these spaces, he restricted himself to subcomplexes, i.e., objects in the space made up only on thesimplices in the Triangulation of the space. Eventually, Poincaré's version of homology was dispensed with and replacedby the more general Singular Homology. Singular Homology is the concept mathematicians mean when they say ``homology.''
In modern usage, however, the word homology is used to mean Homology Group. For example, if someone says ``
did
by computing the homology of 
,'' they mean `` did by computing the Homology Groupsof .'' But sometimes homology is used more loosely in the context of a ``homology in a Space,'' whichcorresponds to singular homology groups.
Singular homology groups of a Space measure the extent to which thereare finite (compact) boundaryless Gadgets in that Space, such that these Gadgetsare not the boundary of other finite (compact) Gadgets in that Space.
A generalized homology or cohomology theory must satisfy all of the Eilenberg-Steenrod Axiomswith the exception of the Dimension Axiom.
See also Cohomology, Dimension Axiom, Eilenberg-Steenrod Axioms, Gadget,Homological Algebra, Homology Group, Simplicial Complex, Simplicial Homology, Singular Homology- Vibration Problem
- Vickery Auction
- Viergruppe
- Vieta's Substitution
- Vigesimal
- Vigintillion
- Villarceau Circles
- Vinculum
- Vinogradov's Theorem
- Virtual Group
- Visibility
- Visible Point
- Visible Point Vector Identity
- Vitali's Convergence Theorem
- Viviani's Curve
- Viviani's Theorem
- Vojta's Conjecture
- Volterra Integral Equation of the First Kind
- Volterra Integral Equation of the Second Kind
- Volume
- Volume Element
- Volume Integral
- von Aubel's Theorem
- von Dyck's Theorem
- von Mangoldt Function