N-connected space

In the mathematical branch of algebraic topology, specifically homotopy theory, n"-connectedness (sometimes, n"-simple connectedness) generalizes the concepts of path-connectedness and simple connectedness. To say that a space is "n"-connected is to say that its first "n" homotopy groups are trivial, and to say that a map is "n"-connected means that it is an isomorphism "up to dimension "n," in homotopy".
"n"-connected space.
A topological space "X" is said to be "n"-connected (for positive "n") when it is non-empty, path-connected, and its first "n" homotopy groups vanish identically, that is
where formula_2 denotes the "i"-th homotopy group and 0 denotes the trivial group.
The requirements of being non-empty and path-connected can be interpreted as (−1)-connected and 0-connected, respectively, which is useful in defining 0-connected and 1-connected maps, as below. 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 "S"0 to the base point of "X". Using this set, a space is 0-connected if and only if the 0th homotopy set is the one-point set. 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 "x"1 and "x"2 in "X" can be connected with a continuous path which starts in "x"1 and ends in "x"2, which is equivalent to the assertion that every mapping from "S"0 (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
"n"-connected map.
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 map formula_5 is "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 "n"th homotopy groups, in the exact sequence:
If the group on the right formula_10 vanishes, then the map on the left is a surjection.
Low-dimensional examples:
"n"-connectivity for spaces can in turn be defined in terms of "n"-connectivity of maps: a space "X" with basepoint "x"0 is an "n"-connected space if and only if the inclusion of the basepoint formula_11 is an "n"-connected map. The 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.
Interpretation.
This is instructive for a subset:
an "n"-connected inclusion formula_12 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 formula_12 to be 1-connected, it must be:
One-to-one on formula_17 means that if there is a path connecting two points formula_18 by passing through "X," there is a path in "A" connecting them, while onto formula_19 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 formula_20 only implies that any elements of formula_21 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 formula_22) 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 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.
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 formula_23 into a more general topological space, such as the space of all continuous maps between two associated spaces formula_24 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.