# 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. The concept of n-connectedness generalizes the concepts of path-connectedness and simple connectedness.
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.
Homotopical connectivity is defined for maps, too. A map is n-connected if it is an isomorphism "up to dimension n, in homotopy".

## Definition using holes

All definitions below consider a topological space X.
A hole in X is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point.$^{:78}$ Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally,
* A d-dimensional sphere in X is a continuous function $f_{d}:S^{d}\to X$.
* A d-dimensional ball in X is a continuous function $g_{d}:B^{d}\to X$.
* A d-dimensional-boundary hole in X is a d-dimensional sphere that is not nullhomotopic (- cannot be shrunk continuously to a point). Equivalently, it is a d-dimensional sphere that cannot be continuously extended to a (d+1)-dimensional ball. It is sometimes called a (d+1)-dimensional hole (d+1 is the dimension of the "missing ball").
* X is called n-connected if it contains no holes of boundary-dimension d ≤ n.$^{:78,Sec.4.3}$
* The homotopical connectivity of X, denoted ${\text{conn}}_{\pi }(X)$, is the largest integer n for which X is n-connected.
* A slightly different definition of connectivity, which makes some computations simpler, is: the smallest integer d such that X contains a d-dimensional hole. This connectivity parameter is denoted by $\eta _{\pi }(X)$, and it differs from the previous parameter by 2, that is, $\eta _{\pi }(X):={\text{conn}}_{\pi }(X)+2$.

### Examples

* A 2-dimensional hole (a hole with a 1-dimensional boundary) is a circle (S$^1$) in X, that cannot be shrunk continuously to a point in X. An example is shown on the figure at the right. The yellow region is the topological space X; it is a pentagon with a triangle removed. The blue circle is a 1-dimensional sphere in X. It cannot be shrunk continuously to a point in X; therefore; X has a 2-dimensional hole. Another example is the punctured plane - the Euclidean plane with a single point removed, $\mathbb {R} ^{2}\setminus \{(0,0)\}$. To make a 2-dimensional hole in a 3-dimensional ball, make a tunnel through it. In general, a space contains a 1-dimensional-boundary hole if and only if it is not simply-connected. Hence, simply-connected is equivalent to 1-connected. X is 0-connected but not 1-connected, so ${\text{conn}}_{\pi }(X)=0$. The lowest dimension of a hole is 2, so $\eta _{\pi }(X)=2$.A 3-dimensional hole.
* A 3-dimensional hole (a hole with a 2-dimensional boundary) is shown on the figure at the right. Here, X is a cube (yellow) with a ball removed (white). The 2-dimensional sphere (blue) cannot be continuously shrunk to a single point. X is simply-connected but not 2-connected, so ${\text{conn}}_{\pi }(X)=1$. The smallest dimension of a hole is 3, so $\eta _{\pi }(X)=3$.
* For a 1-dimensional hole (a hole with a 0-dimensional boundary) we need to consider $S^{0}$ - the zero-dimensional sphere. What is a zero dimensional sphere? - For every integer d, the sphere $S^{d}$ is the boundary of the (d+1)-dimensional ball $B^{d+1}$. So $S^{0}$ is the boundary of $B^{1}$, which is the segment [0,1]. Therefore, $S^{0}$ is the set of two disjoint points {0, 1}. A zero-dimensional sphere in X is just a set of two points in X. If there is such a set, that cannot be continuously shrunk to a single point in X (or continuously extended to a segment in X), this means that there is no path between the two points, that is, X is not path-connected; see the figure at the right. Hence, path-connected is equivalent to 0-connected. X is not 0-connected, so ${\text{conn}}_{\pi }(X)=-1$. The lowest dimension of a hole is 1, so $\eta _{\pi }(X)=1$.
* A 0-dimensional hole is a missing 0-dimensional ball. A 0-dimensional ball is a single point; its boundary $S^{-1}$ is an empty set. Therefore, the existence of a 0-dimensional hole is equivalent to the space being empty. Hence, non-empty is equivalent to (-1)-connected. For an empty space X, ${\text{conn}}_{\pi }(X)=-2$ and $\eta _{\pi }(X)=0$, which is its smallest possible value.
* A ball has no holes of any dimension. Therefore, its connectivity is infinite: $\eta _{\pi }(X)={\text{conn}}_{\pi }(X)=\infty$.

### Homotopical connectivity of spheres

In general, for every integer d, ${\text{conn}}_{\pi }(S^{d})=d-1$ (and $\eta _{\pi }(S^{d})=d+1$)$^{:79,Thm.4.3.2}$ The proof requires two directions:
* Proving that ${\text{conn}}_{\pi }(S^{d})<d$, that is, $S^{d}$ cannot be continuously shrunk to a single point. This can be proved using the Borsuk–Ulam theorem.
* Proving that ${\text{conn}}_{\pi }(S^{d})\geq d-1$, that is, that is, every continuous map $S^{k}\to S^{d}$ for $k<d$ can be continuously shrunk to a single point.

## Definition using groups

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: $\pi _{d}(X)\cong 0,\quad -1\leq d\leq n,$ where $\pi _{i}(X)$ denotes the i-th homotopy group and 0 denotes the trivial group. The two definitions are equivalent. The requirement for an n-connected space consists of requirements for all d ≤ n:
* The requirement for d=-1 means that X should be nonempty.
* The requirement for d=0 means that X should be path-connected.
* The requirement for any d ≥ 1 means that X contains no holes of boundary dimension d. That is, every d-dimensional sphere in X is homotopic to a constant map. Therefore, the d-th homotopy group of X is trivial. The opposite is also true: If X has a hole with a d-dimensional boundary, then there is a d-dimensional sphere that is not homotopic to a constant map, so the d-th homotopy group of X is not trivial. In short, X has a hole with a d-dimensional boundary, if-and-only-if $\pi _{d}(X)\not \cong 0$.The homotopical connectivity of X is the largest integer n for which X is n-connected.
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:
    $\pi _{0}(X,*):=\left[\left(S^{0},*\right),\left(X,*\right)\right].$
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
    $\pi _{i}(X)\simeq 0,\quad 0\leq i\leq n.$

### 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:
* $\pi _{i}(f)\colon \pi _{i}(X)\mathrel {\overset {\sim }{\to }} \pi _{i}(Y)$ 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 nth homotopy groups in the exact sequence
    $\pi _{n}(X)\mathrel {\overset {\pi _{n}(f)}{\to }} \pi _{n}(Y)\to \pi _{n-1}(Ff).$
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 x$_0$ 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 elements 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 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.

## Lower bounds

Many topological proofs require lower bounds on the homotopical connectivity. There are several "recipes" for proving such lower bounds.

### Homology

The Hurewicz theorem relates the homotopical connectivity ${\text{conn}}_{\pi }(X)$ to the homological connectivity, denoted by ${\text{conn}}_{H}(X)$. This is useful for computing homotopical connectivity, since the homological groups can be computed more easily.
Suppose first that X is simply-connected, that is, ${\text{conn}}_{\pi }(X)\geq 1$. Let $n:={\text{conn}}_{\pi }(X)+1\geq 2$; so $\pi _{i}(X)=0$ for all $i<n$, and $\pi _{n}(X)\neq 0$. Hurewicz theorem$^{:366,Thm.4.32}$ says that, in this case, ${\tilde {H_{i}}}(X)=0$ for all $i<n$, and ${\tilde {H_{n}}}(X)$ is isomorphic to $\pi _{n}(X)$, so ${\tilde {H_{n}}}(X)\neq 0$ too. Therefore:${\text{conn}}_{H}(X)={\text{conn}}_{\pi }(X).$If X is not simply-connected (${\text{conn}}_{\pi }(X)\leq 0$), then${\text{conn}}_{H}(X)\geq {\text{conn}}_{\pi }(X)$still holds. When ${\text{conn}}_{\pi }(X)\leq -1$ this is trivial. When ${\text{conn}}_{\pi }(X)=0$ (so X is path-connected but not simply-connected), one should prove that ${\tilde {H_{0}}}(X)=0$.
The inequality may be strict: there are spaces in which ${\text{conn}}_{\pi }(X)=0$ but ${\text{conn}}_{H}(X)=\infty$.
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.$^{:80,Prop.4.4.2}$

### Join

Let K and L be non-empty cell complexes. Their join is commonly denoted by $K*L$. Then:$^{:81,Prop.4.4.3}$ ${\text{conn}}_{\pi }(K*L)\geq {\text{conn}}_{\pi }(K)+{\text{conn}}_{\pi }(L)+2.$
The identity is simpler with the eta notation: $\eta _{\pi }(K*L)\geq \eta _{\pi }(K)+\eta _{\pi }(L).$ As an example, let $K=L=S^{0}=$ a set of two disconnected points. There is a 1-dimensional hole between the points, so the eta is 1. The join $K*L$ 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 to $S^{2}$, and its eta is 3. In general, the join of n copies of $S^{0}$ is homeomorphic to $S^{n-1}$ and its eta is n.
The general proof is based on a similar formula for the homological connectivity.

### Nerve

Let K$_1$,...,K$_n$ be abstract simplicial complexes, and denote their union by K.
Denote the nerve complex of {K$_1$, ... , K$_n$} (the abstract complex recording the intersection pattern of the K$_i$) by N.
If, for each nonempty $J\subset I$, the intersection ⋂$_{i∈J}$U$_i${\textstyle \bigcap _{i\in J}U_{i}} is 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.$^{:Thm.6}$

## Homotopy principle

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.