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Concept
Homotopical connectivity
Summary
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness.
An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is n-connected (or n-simple connected) if its first n homotopy groups are trivial.
Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".
All definitions below consider a topological space X.
A hole in X is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point. Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally,
A d-dimensional sphere in X is a continuous function .
A d-dimensional ball in X is a continuous function .
A d-dimensional-boundary hole in X is a d-dimensional sphere that is not nullhomotopic (- cannot be shrunk continuously to a point). Equivalently, it is a d-dimensional sphere that cannot be continuously extended to a (d+1)-dimensional ball. It is sometimes called a (d+1)-dimensional hole (d+1 is the dimension of the "missing ball").
X is called n-connected if it contains no holes of boundary-dimension d ≤ n.
The homotopical connectivity of X, denoted , is the largest integer n for which X is n-connected.
A slightly different definition of connectivity, which makes some computations simpler, is: the smallest integer d such that X contains a d-dimensional hole. This connectivity parameter is denoted by , and it differs from the previous parameter by 2, that is, .
A 2-dimensional hole (a hole with a 1-dimensional boundary) is a circle (S1) in X, that cannot be shrunk continuously to a point in X. An example is shown on the figure at the right. The yellow region is the topological space X; it is a pentagon with a triangle removed.
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