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".
Definition using holes 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 f d : S d → X {\displaystyle f_{d}:S^{d}\to X} . A d-dimensional ball in X is a continuous function g d : B d → X {\displaystyle g_{d}:B^{d}\to X} . 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 conn π ( X ) {\displaystyle {\text{conn}}_{\pi }(X)} , 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 η π ( X ) {\displaystyle \eta _{\pi }(X)} , and it differs from the previous parameter by 2, that is, η π ( X ) := conn π ( X ) + 2 {\displaystyle \eta _{\pi }(X):={\text{conn}}_{\pi }(X)+2} .
Examples
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. The blue circle is a 1-dimensional sphere in X. It cannot be shrunk continuously to a point in X; therefore; X has a 2-dimensional hole. Another example is the punctured plane - the Euclidean plane with a single point removed, R 2 ∖ { ( 0 , 0 ) } {\displaystyle \mathbb {R} ^{2}\setminus \{(0,0)\}} . To make a 2-dimensional hole in a 3-dimensional ball, make a tunnel through it. In general, a space contains a 1-dimensional-boundary hole if and only if it is not simply-connected. Hence, simply-connected is equivalent to 1-connected. X is 0-connected but not 1-connected, so conn π ( X ) = 0 {\displaystyle {\text{conn}}_{\pi }(X)=0} . The lowest dimension of a hole is 2, so η π ( X ) = 2 {\displaystyle \eta _{\pi }(X)=2} . A 3-dimensional hole (a hole with a 2-dimensional boundary) is shown on the figure at the right. Here, X is a cube (yellow) with a ball removed (white). The 2-dimensional sphere (blue) cannot be continuously shrunk to a single point. X is simply-connected but not 2-connected, so conn π ( X ) = 1 {\displaystyle {\text{conn}}_{\pi }(X)=1} . The smallest dimension of a hole is 3, so η π ( X ) = 3 {\displaystyle \eta _{\pi }(X)=3} .
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