- n-connected
-
This article is about the concept in algebraic topology. For the concept in graph theory, see Connectivity (graph theory).
In the mathematical branch of algebraic topology, specifically homotopy theory, n-connectedness is a way to say that a space vanishes or that a map is an isomorphism "up to dimension n, in homotopy".
Contents
n-connected space
A topological space X is said to be n-connected if and only if it is path-connected and its first n homotopy groups vanish identically, that is
where the left-hand side denotes the i-th homotopy group. The requirement of being path-connected can also be expressed as 0-connectedness, when defining the 0th homotopy set as:
- π0(X, * ): = [(S0, * ),(X, * )];
this is only a pointed set, not a group, unless X is itself a topological group.
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
Examples
- As described above, a space X is 0-connected if and only if it is path-connected.
- A space is 1-connected if and only if it is simply connected. Thus, the term n-connected is a natural generalization of being path-connected or simply connected.
It is obvious from the definition that an n-connected space X is also i-connected for all i<n.
n-connected map
The corresponding relative notion to the absolute notion of an n-connected space is an n-connected map, which is almost defined as a map whose homotopy cofiber Cf is an n-connected space. In terms of homotopy groups, it means that a map is n-connected if and only if:
- is an isomorphism for i < n, and
- is a surjection.
The last condition is frequently confusing; it is because the vanishing of the nth homotopy of the homotopy cofiber Cf corresponds surjection on the nth homotopy groups, in the exact sequence:
If the group on the right πn(Cf) vanishes, then the map on the left is a surjection.
For instance, a simply connected map (1-connected map) is one that is an isomorphism on path-components, and onto the fundamental group.
Interpretation
This is instructive for a subset: an n-connected inclusion is one such that, up to dimension n−1, homotopies in the larger space X can be homotoped into homotopies in the subset A.
For example, for an inclusion map to be 1-connected, it must be:
- onto π0(X),
- one-to-one on and
- onto π1(X).
One-to-one on means that if there is a path connecting two points by passing through X, there is a path in A connecting them, while onto π1(X) means that in fact a path in X is homotopic to a path in A.
In other words, a function which is an isomorphism on only implies that any element of πn − 1(A) that are homotopic in X are abstractly homotopic in A – the homotopy in A may be unrelated to the homotopy in X – while being n-connected (so also onto πn(X)) means that (up to dimension n−1) homotopies in X can be pushed into homotopies in A.
This gives a more concrete explanation for the utility of the definition of n-connectedness: for example, a space such that the inclusion of the k-skeleton in n-connected (for n>k) – such as the inclusion of a point in the n-sphere – means that any cells in dimension between k and n are not affecting the homotopy type from the point of view of low dimensions.
Applications
The concept of n-connectedness is used in the Hurewicz theorem which describes the relation between singular homology and the higher homotopy groups.
In geometric topology, cases when the inclusion of a geometrically-defined space, such as the space of immersions into a more general topological space, such as the space of all continuous maps between two associated spaces 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.
See also
- connected space
- simply connected
- path-connected
- homotopy group
Categories:- Connection (mathematics)
- General topology
- Properties of topological spaces
Wikimedia Foundation. 2010.