Homotopical connectivity

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.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.A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".A hole in X is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point.[1]: 78 Equivalently, it is a sphere that cannot be continuously extended to a ball.)[1]: 79, Thm.4.3.2 The proof requires two directions: A space X is called n-connected, for n ≥ 0, if it is non-empty, and all its homotopy groups of order d ≤ n are the trivial group:The 0th homotopy set can be defined as: This is only a pointed set, not a group, unless X is itself a topological group; the distinguished point is the class of the trivial map, sending S0 to the base point of X.The definition of homotopy groups and this homotopy set require that X be pointed (have a chosen base point), which cannot be done if X is empty.A topological space X is path-connected if and only if its 0th homotopy group vanishes identically, as path-connectedness implies that any two points x1 and x2 in X can be connected with a continuous path which starts in x1 and ends in x2, which is equivalent to the assertion that every mapping from S0 (a discrete set of two points) to X can be deformed continuously to a constant map.With this definition, we can define X to be n-connected if and only if The corresponding relative notion to the absolute notion of an n-connected space is an n-connected map, which is defined as a map whose homotopy fiber Ff is an (n − 1)-connected space.In terms of homotopy groups, it means that a mapis n-connected if and only if: The last condition is frequently confusing; it is because the vanishing of the (n − 1)-st homotopy group of the homotopy fiber Ff corresponds to a surjection on the nth homotopy groups in the exact sequence If the group on the rightLow-dimensional examples: n-connectivity for spaces can in turn be defined in terms of n-connectivity of maps: a space X with basepoint x0 is an n-connected space if and only if the inclusion of the basepointThe single point set is contractible, so all its homotopy groups vanish, and thus "isomorphism below n and onto at n" corresponds to the first n homotopy groups of X vanishing.This gives a more concrete explanation for the utility of the definition of n-connectedness: for example, a space where the inclusion of the k-skeleton is n-connected (for n > k) – such as the inclusion of a point in the n-sphere – has the property that any cells in dimensions between k and n do not affect the lower-dimensional homotopy types.Many topological proofs require lower bounds on the homotopical connectivity.There are several "recipes" for proving such lower bounds.The Hurewicz theorem relates the homotopical connectivity[clarification needed] The inequality may be strict: there are spaces in which[6] By definition, the k-th homology group of a simplicial complex depends only on the simplices of dimension at most k+1 (see simplicial homology).Therefore, the above theorem implies that a simplicial complex K is k-connected if and only if its (k+1)-dimensional skeleton (the subset of K containing only simplices of dimension at most k+1) is k-connected.is a square, which is homeomorphic to a circle, so its eta is 2.The join of this square with a third copy of K is a octahedron, which is homeomorphic toand its eta is n. The general proof is based on a similar formula for the homological connectivity.Let K1,...,Kn be abstract simplicial complexes, and denote their union by K. Denote the nerve complex of {K1, ... , Kn} (the abstract complex recording the intersection pattern of the Ki) by N. If, for each nonemptyis either empty or (k−|J|+1)-connected, then for every j ≤ k, the j-th homotopy group of N is isomorphic to the j-th homotopy group of K. In particular, N is k-connected if-and-only-if K is k-connected.are n-connected are said to satisfy a homotopy principle or "h-principle".There are a number of powerful general techniques for proving h-principles.