n-connected space

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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

${\displaystyle \pi _{i}(X)\cong 0,\quad 1\leq i\leq n,}$

where ${\displaystyle \pi _{i}(X)}$  denotes the i-th homotopy group and 0 denotes the trivial group.[1]

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 0-th homotopy set can be defined as:

${\displaystyle \pi _{0}(X,*):=[(S^{0},*),(X,*)].}$

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. 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 0-th 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

${\displaystyle \pi _{i}(X)\simeq 0,\quad 0\leq i\leq n.}$

Examples

• A space X is (−1)-connected if and only if it is non-empty.
• A space X is 0-connected if and only if it is non-empty and path-connected.
• A space is 1-connected if and only if it is simply connected.
• An n-sphere is (n − 1)-connected.

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 ${\displaystyle f\colon X\to Y}$  is n-connected if and only if:

• ${\displaystyle \pi _{i}(f)\colon \pi _{i}(X){\overset {\sim }{\to }}\pi _{i}(Y)}$  is an isomorphism for ${\displaystyle i , and
• ${\displaystyle \pi _{n}(f)\colon \pi _{n}(X)\twoheadrightarrow \pi _{n}(Y)}$  is a surjection.

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:

${\displaystyle \pi _{n}(X){\overset {\pi _{n}(f)}{\to }}\pi _{n}(Y)\to \pi _{n-1}(Ff).}$

If the group on the right ${\displaystyle \pi _{n-1}(Ff)}$  vanishes, then the map on the left is a surjection.

Low-dimensional examples:

• A connected map (0-connected map) is one that is onto path components (0th homotopy group); this corresponds to the homotopy fiber being non-empty.
• A simply connected map (1-connected map) is one that is an isomorphism on path components (0th homotopy group) and onto the fundamental group (1st homotopy group).

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 basepoint ${\displaystyle x_{0}\hookrightarrow X}$  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 ${\displaystyle A\hookrightarrow X}$  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 ${\displaystyle A\hookrightarrow X}$  to be 1-connected, it must be:

• onto ${\displaystyle \pi _{0}(X),}$
• one-to-one on ${\displaystyle \pi _{0}(A)\to \pi _{0}(X),}$  and
• onto ${\displaystyle \pi _{1}(X).}$

One-to-one on ${\displaystyle \pi _{0}(A)\to \pi _{0}(X)}$  means that if there is a path connecting two points ${\displaystyle a,b\in A}$  by passing through X, there is a path in A connecting them, while onto ${\displaystyle \pi _{1}(X)}$  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 ${\displaystyle \pi _{n-1}(A)\to \pi _{n-1}(X)}$  only implies that any elements of ${\displaystyle \pi _{n-1}(A)}$  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 ${\displaystyle \pi _{n}(X)}$ ) 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 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 ${\displaystyle M\to N,}$  into a more general topological space, such as the space of all continuous maps between two associated spaces ${\displaystyle X(M)\to X(N),}$  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.