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".

## Contents

* 1 Definition using holes
* 1.1 Examples
* 1.2 Homotopical connectivity of spheres
* 2 Definition using groups
* 2.1 Examples
* 3 n-connected map
* 3.1 Interpretation
* 4 Lower bounds
* 4.1 Homology
* 4.2 Join
* 4.3 Nerve
* 5 Homotopy principle
* 6 See also
* 7 References

## 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.[1](p78) Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally,

* A d-dimensional sphere in X is a continuous function [math]\displaystyle{ f_d: S^d \to X }[/math].
* A d-dimensional ball in X is a continuous function [math]\displaystyle{ g_d: B^d \to X }[/math].
* 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](p78)
* The homotopical connectivity of X, denoted [math]\displaystyle{ \text{conn}_{\pi}(X) }[/math], 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 [math]\displaystyle{ \eta_{\pi}(X) }[/math], and it differs from the previous parameter by 2, that is, [math]\displaystyle{ \eta_{\pi}(X) := \text{conn}_{\pi}(X) + 2 }[/math].[2]

### Examples

A 2-dimensional hole (a hole with a 1-dimensional boundary).

* 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, [math]\displaystyle{ \mathbb{R}^2\setminus \\{(0,0)\\} }[/math]. 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 [math]\displaystyle{ \text{conn}_{\pi}(X) = 0 }[/math]. The lowest dimension of a hole is 2, so [math]\displaystyle{ \eta_{\pi}(X) = 2 }[/math].

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 [math]\displaystyle{ \text{conn}_{\pi}(X) = 1 }[/math]. The smallest dimension of a hole is 3, so [math]\displaystyle{ \eta_{\pi}(X) = 3 }[/math].

A 1-dimensional hole.

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

### Homotopical connectivity of spheres

In general, for every integer d, [math]\displaystyle{ \text{conn}_{\pi}(S^d)=d-1 }[/math] (and [math]\displaystyle{ \eta_{\pi}(S^d)=d+1 }[/math])[1](p79) The proof requires two directions:

* Proving that [math]\displaystyle{ \text{conn}_{\pi}(S^d) \lt d }[/math], that is, [math]\displaystyle{ S^d }[/math] cannot be continuously shrunk to a single point. This can be proved using the Borsuk–Ulam theorem.
* Proving that [math]\displaystyle{ \text{conn}_{\pi}(S^d) \geq d-1 }[/math], that is, that is, every continuous map [math]\displaystyle{ S^k \to S^d }[/math] for [math]\displaystyle{ k \lt d }[/math] 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: [math]\displaystyle{ \pi_d(X) \cong 0, \quad -1 \leq d \leq n, }[/math] where [math]\displaystyle{ \pi_i(X) }[/math] 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 [math]\displaystyle{ \pi_d(X) \not \cong 0 }[/math].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:

[math]\displaystyle{ \pi_0(X, *) := \left[\left(S^0, *\right), \left(X, *\right)\right]. }[/math]

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

[math]\displaystyle{ \pi_i(X) \simeq 0, \quad 0 \leq i \leq n. }[/math]

### 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 [math]\displaystyle{ f\colon X \to Y }[/math] is n-connected if and only if:

* [math]\displaystyle{ \pi_i(f)\colon \pi_i(X) \mathrel{\overset{\sim}{\to}} \pi_i(Y) }[/math] is an isomorphism for [math]\displaystyle{ i \lt n }[/math], and
* [math]\displaystyle{ \pi_n(f)\colon \pi_n(X) \twoheadrightarrow \pi_n(Y) }[/math] 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:

[math]\displaystyle{ \pi_n(X) \mathrel{\overset{\pi_n(f)}{\to}} \pi_n(Y) \to \pi_{n-1}(Ff). }[/math]

If the group on the right [math]\displaystyle{ \pi_{n-1}(Ff) }[/math] 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 [math]\displaystyle{ x_0 \hookrightarrow X }[/math] 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 [math]\displaystyle{ A \hookrightarrow X }[/math] 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 [math]\displaystyle{ A \hookrightarrow X }[/math] to be 1-connected, it must be:

* onto [math]\displaystyle{ \pi_0(X), }[/math]
* one-to-one on [math]\displaystyle{ \pi_0(A) \to \pi_0(X), }[/math] and
* onto [math]\displaystyle{ \pi_1(X). }[/math]

One-to-one on [math]\displaystyle{ \pi_0(A) \to \pi_0(X) }[/math] means that if there is a path connecting two points [math]\displaystyle{ a, b \in A }[/math] by passing through X, there is a path in A connecting them, while onto [math]\displaystyle{ \pi_1(X) }[/math] 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 [math]\displaystyle{ \pi_{n-1}(A) \to \pi_{n-1}(X) }[/math] only implies that any elements of [math]\displaystyle{ \pi_{n-1}(A) }[/math] 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 [math]\displaystyle{ \pi_n(X) }[/math]) 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

Hurewicz theorem relates the homotopical connectivity [math]\displaystyle{ \text{conn}_{\pi}(X) }[/math] to the homological connectivity, denoted by [math]\displaystyle{ \text{conn}_H(X) }[/math]. 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, [math]\displaystyle{ \text{conn}_{\pi}(X)\geq 1 }[/math]. Let [math]\displaystyle{ n := \text{conn}_{\pi}(X) + 1\geq 2 }[/math]; so [math]\displaystyle{ \pi_i(X)= 0 }[/math] for all [math]\displaystyle{ i\lt n }[/math], and [math]\displaystyle{ \pi_n(X)\neq 0 }[/math]. Hurewicz theorem[5](p366) says that, in this case, [math]\displaystyle{ \tilde{H_i}(X)= 0 }[/math] for all [math]\displaystyle{ i\lt n }[/math], and [math]\displaystyle{ \tilde{H_n}(X) }[/math] is isomorphic to [math]\displaystyle{ \pi_n(X) }[/math], so [math]\displaystyle{ \tilde{H_n}(X)\neq 0 }[/math] too. Therefore:[math]\displaystyle{ \text{conn}_H(X) = \text{conn}_{\pi}(X). }[/math]If X is not simply-connected ([math]\displaystyle{ \text{conn}_{\pi}(X)\leq 0 }[/math]), then[math]\displaystyle{ \text{conn}_H(X)\geq \text{conn}_{\pi}(X) }[/math]still holds. When [math]\displaystyle{ \text{conn}_{\pi}(X)\leq-1 }[/math] this is trivial. When [math]\displaystyle{ \text{conn}_{\pi}(X)=0 }[/math] (so X is path-connected but not simply-connected), one should prove that [math]\displaystyle{ \tilde{H_0}(X)= 0 }[/math].[clarification needed]

The inequality may be strict: there are spaces in which [math]\displaystyle{ \text{conn}_{\pi}(X)=0 }[/math] but [math]\displaystyle{ \text{conn}_H(X)=\infty }[/math].[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](p80)

### Join

Let K and L be non-empty cell complexes. Their join is commonly denoted by [math]\displaystyle{ K * L }[/math]. Then:[1](p81) [math]\displaystyle{ \text{conn}_{\pi}(K*L) \geq \text{conn}_{\pi}(K)+\text{conn}_{\pi}(L)+2. }[/math]

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

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

### Nerve

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

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

If, for each nonempty [math]\displaystyle{ J\subset I }[/math], the intersection [math]\displaystyle{ \bigcap_{i\in J} U_i }[/math] 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]:{{{1}}}

## Homotopy principle

In geometric topology, cases when the inclusion of a geometrically-defined space, such as the space of immersions [math]\displaystyle{ M \to N, }[/math] into a more general topological space, such as the space of all continuous maps between two associated spaces [math]\displaystyle{ X(M) \to X(N), }[/math] 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

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Template:Cite Matousek 2007, Section 4.3
2. ↑ Aharoni, Ron; Berger, Eli (2006). "The intersection of a matroid and a simplicial complex" (in en). Transactions of the American Mathematical Society 358 (11): 4895–4917. doi:10.1090/S0002-9947-06-03833-5. ISSN 0002-9947. https://www.ams.org/tran/2006-358-11/S0002-9947-06-03833-5/.
3. ↑ "n-connected space in nLab". https://ncatlab.org/nlab/show/n-connected+space.
4. ↑ Frick, Florian; Soberón, Pablo (2020-05-11). "The topological Tverberg problem beyond prime powers". arXiv:2005.05251 [math.CO].
5. ↑ Hatcher, Allen (2001), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79160-1
6. ↑ See example 2.38 in Hatcher's book. See also this answer.
7. ↑ Björner, Anders (2003-04-01). "Nerves, fibers and homotopy groups" (in en). Journal of Combinatorial Theory. Series A 102 (1): 88–93. doi:10.1016/S0097-3165(03)00015-3. ISSN 0097-3165.

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