Hole

A hole in a mathematical object is a Topological structure which prevents the object from beingcontinuously shrunk to a point. When dealing with Topological Spaces, aDisconnectivity is interpreted as a hole in the space. Examples of holes are things like the hole in the``center'' of a Sphere or a Circle and the hole produced in Euclidean Space cutting a Knotout from it.

Singular Homology Groups form a Measure of the hole structure of a Space, but theyare one particular measure and they don't always pick up everything. Homotopy Groups of aSpace are another measure of holes in a Space, as well as Bordism Groups,k-Theory, Cohomotopy Groups, and so on.

There are many ways to measure holes in a space. Some holes are picked up by Homotopy Groups that arenot picked up by Homology Groups, and some holes are picked up by Homology Groups that are not picked up by Homotopy Groups. (For example, in the Torus,Homotopy Groups ``miss'' the two-dimensional hole that is given by the Torus itself, but thesecond Homology Group picks that hole up.) In addition, Homology Groups don't pick up thevarying hole structures of the complement of Knots in 3-space, but the first Homotopy Group (thefundamental group) does.

• Minimal Cover
• Minimal Discriminant
• Minimal Matrix
• Minimal Residue
• Minimal Set
• Minimal Surface
• Minimax Approximation
• Minimax Polynomial
• Minimax Theorem
• Minimum
• Minkowski-Bouligand Dimension
• Minkowski Convex Body Theorem
• Minkowski Geometry
• Minkowski-Hlawka Theorem
• Minkowski Integral Inequality
• Minkowski Measure
• Minkowski Metric
• Minkowski Sausage
• Minkowski Space
• Minkowski Sum
• Minkowski Sum Inequality
• Minor
• Minor Axis
• Minor Graph
• Minus