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".

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.^[1]: 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 ${\displaystyle f_{d}:S^{d}\to X}$.
• A d-dimensional ball in X is a continuous function ${\displaystyle g_{d}:B^{d}\to X}$.
• 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.^{[1]: 78, Sec.4.3}
• The homotopical connectivity of X, denoted ${\displaystyle {\text{conn}}_{\pi }(X)}$, 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 ${\displaystyle \eta _{\pi }(X)}$, and it differs from the previous parameter by 2, that is, ${\displaystyle \eta _{\pi }(X):={\text{conn}}_{\pi }(X)+2}$.^[2]

• A 2-dimensional hole (a hole with a 1-dimensional boundary) is a circle (S¹) 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, ${\displaystyle \mathbb {R} ^{2}\setminus \{(0,0)\}}$. To make a 2-dimensional hole in a 3-dimensional ball, make a tunnel through it.^[1] 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 ${\displaystyle {\text{conn}}_{\pi }(X)=0}$. The lowest dimension of a hole is 2, so ${\displaystyle \eta _{\pi }(X)=2}$.

  

• 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 ${\displaystyle {\text{conn}}_{\pi }(X)=1}$. The smallest dimension of a hole is 3, so ${\displaystyle \eta _{\pi }(X)=3}$.

• For a 1-dimensional hole (a hole with a 0-dimensional boundary) we need to consider ${\displaystyle S^{0}}$ - the zero-dimensional sphere. What is a zero dimensional sphere? - For every integer d, the sphere ${\displaystyle S^{d}}$ is the boundary of the (d+1)-dimensional ball ${\displaystyle B^{d+1}}$. So ${\displaystyle S^{0}}$ is the boundary of ${\displaystyle B^{1}}$, which is the segment [0,1]. Therefore, ${\displaystyle S^{0}}$ 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 ${\displaystyle {\text{conn}}_{\pi }(X)=-1}$. The lowest dimension of a hole is 1, so ${\displaystyle \eta _{\pi }(X)=1}$.
• A 0-dimensional hole is a missing 0-dimensional ball. A 0-dimensional ball is a single point; its boundary ${\displaystyle S^{-1}}$ 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, ${\displaystyle {\text{conn}}_{\pi }(X)=-2}$ and ${\displaystyle \eta _{\pi }(X)=0}$, which is its smallest possible value.
• A ball has no holes of any dimension. Therefore, its connectivity is infinite: ${\displaystyle \eta _{\pi }(X)={\text{conn}}_{\pi }(X)=\infty }$.

In general, for every integer d, ${\displaystyle {\text{conn}}_{\pi }(S^{d})=d-1}$ (and ${\displaystyle \eta _{\pi }(S^{d})=d+1}$)^{[1]: 79, Thm.4.3.2} The proof requires two directions:

• Proving that ${\displaystyle {\text{conn}}_{\pi }(S^{d})<d}$, that is, ${\displaystyle S^{d}}$ cannot be continuously shrunk to a single point. This can be proved using the Borsuk–Ulam theorem.
• Proving that ${\displaystyle {\text{conn}}_{\pi }(S^{d})\geq d-1}$, that is, that is, every continuous map ${\displaystyle S^{k}\to S^{d}}$ for ${\displaystyle k<d}$ can be continuously shrunk to a single point.

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: $${\displaystyle \pi _{d}(X)\cong 0,\quad -1\leq d\leq n,}$$ where ${\displaystyle \pi _{i}(X)}$ denotes the i-th homotopy group and 0 denotes the trivial group.^[3] 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 ${\displaystyle \pi _{d}(X)\not \cong 0}$.The homotopical connectivity of X is the largest integer n for which X is n-connected.^[4]

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:

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

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 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 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

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

• 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.

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:

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

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

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.

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.

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

The Hurewicz theorem relates the homotopical connectivity ${\displaystyle {\text{conn}}_{\pi }(X)}$ to the homological connectivity, denoted by ${\displaystyle {\text{conn}}_{H}(X)}$. 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, ${\displaystyle {\text{conn}}_{\pi }(X)\geq 1}$. Let ${\displaystyle n:={\text{conn}}_{\pi }(X)+1\geq 2}$; so ${\displaystyle \pi _{i}(X)=0}$ for all ${\displaystyle i<n}$, and ${\displaystyle \pi _{n}(X)\neq 0}$. Hurewicz theorem^{[5]: 366, Thm.4.32} says that, in this case, ${\displaystyle {\tilde {H_{i}}}(X)=0}$ for all ${\displaystyle i<n}$, and ${\displaystyle {\tilde {H_{n}}}(X)}$ is isomorphic to ${\displaystyle \pi _{n}(X)}$, so ${\displaystyle {\tilde {H_{n}}}(X)\neq 0}$ too. Therefore:$${\displaystyle {\text{conn}}_{H}(X)={\text{conn}}_{\pi }(X).}$$If X is not simply-connected (${\displaystyle {\text{conn}}_{\pi }(X)\leq 0}$), then$${\displaystyle {\text{conn}}_{H}(X)\geq {\text{conn}}_{\pi }(X)}$$still holds. When ${\displaystyle {\text{conn}}_{\pi }(X)\leq -1}$ this is trivial. When ${\displaystyle {\text{conn}}_{\pi }(X)=0}$ (so X is path-connected but not simply-connected), one should prove that ${\displaystyle {\tilde {H_{0}}}(X)=0}$.^{[clarification needed]}

The inequality may be strict: there are spaces in which ${\displaystyle {\text{conn}}_{\pi }(X)=0}$ but ${\displaystyle {\text{conn}}_{H}(X)=\infty }$.^[6]

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.^{[1]: 80, Prop.4.4.2}

Let K and L be non-empty cell complexes. Their join is commonly denoted by ${\displaystyle K*L}$. Then:^{[1]: 81, Prop.4.4.3} $${\displaystyle {\text{conn}}_{\pi }(K*L)\geq {\text{conn}}_{\pi }(K)+{\text{conn}}_{\pi }(L)+2.}$$

The identity is simpler with the eta notation: $${\displaystyle \eta _{\pi }(K*L)\geq \eta _{\pi }(K)+\eta _{\pi }(L).}$$ As an example, let ${\displaystyle K=L=S^{0}=}$ a set of two disconnected points. There is a 1-dimensional hole between the points, so the eta is 1. The join ${\displaystyle K*L}$ 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 ${\displaystyle S^{2}}$, and its eta is 3. In general, the join of n copies of ${\displaystyle S^{0}}$ is homeomorphic to ${\displaystyle S^{n-1}}$ and its eta is n.

The general proof is based on a similar formula for the homological connectivity.

Let K₁,...,K_n be abstract simplicial complexes, and denote their union by K.

Denote the nerve complex of {K₁, ... , K_n} (the abstract complex recording the intersection pattern of the K_i) by N.

If, for each nonempty ${\displaystyle J\subset I}$, the intersection ${\textstyle \bigcap _{i\in J}U_{i}}$ 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.^[7]: Thm.6

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.

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