Homotopical connectivity

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 (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

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

where $\pi _{i}(X)$ 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 0-th homotopy set can be defined as:

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

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

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

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

• $\pi _{i}(f)\colon \pi _{i}(X){\overset {\sim }{\to }}\pi _{i}(Y)$ is an isomorphism for $i<n$, and
• $\pi _{n}(f)\colon \pi _{n}(X)\twoheadrightarrow \pi _{n}(Y)$ 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 n^th homotopy groups, in the exact sequence:

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

If the group on the right $\pi _{n-1}(Ff)$ 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 $x_{0}\hookrightarrow X$ 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 $A\hookrightarrow X$ 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 $A\hookrightarrow X$ to be 1-connected, it must be:

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

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

See also

• Connected space
• Connective spectrum
• Path-connected
• Simply connected