Homotopical connectivity

Not to be confused with Homotopic connectivity.

This article is about the concept in algebraic topology. For other connectivity concepts, see 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

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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 f d : S d → X {\displaystyle f_{d}:S^{d}\to X} .
* A d-dimensional ball in X is a continuous function  g d : B d → X {\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  conn π ( X ) {\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  η π ( X ) {\displaystyle \eta _{\pi }(X)} , and it differs from the previous parameter by 2, that is,  η π ( X ) := conn π ( X ) + 2 {\displaystyle \eta _{\pi }(X):={\text{conn}}_{\pi }(X)+2} .[2]

### Examples

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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,  R 2 ∖ { ( 0 , 0 ) } {\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  conn π ( X ) = 0 {\displaystyle {\text{conn}}_{\pi }(X)=0} . The lowest dimension of a hole is 2, so  η π ( X ) = 2 {\displaystyle \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  conn π ( X ) = 1 {\displaystyle {\text{conn}}_{\pi }(X)=1} . The smallest dimension of a hole is 3, so  η π ( X ) = 3 {\displaystyle \eta _{\pi }(X)=3} .

A 1-dimensional hole.

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

### Homotopical connectivity of spheres

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

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

## Definition using groups

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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:  π d ( X ) ≅ 0 , − 1 ≤ d ≤ n , {\displaystyle \pi _{d}(X)\cong 0,\quad -1\leq d\leq n,} where  π i ( X ) {\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  π d ( X ) ≇ 0 {\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:

π 0 ( X , ∗ ) := [ ( S 0 , ∗ ) , ( X , ∗ ) ] . {\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

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

### Examples

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* 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

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

* π i ( f ) : π i ( X ) → ∼ π i ( Y ) {\displaystyle \pi _{i}(f)\colon \pi _{i}(X)\mathrel {\overset {\sim }{\to }} \pi _{i}(Y)} is an isomorphism for  i < n {\displaystyle i<n} , and
* π n ( f ) : π n ( X ) ↠ π n ( Y ) {\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

π n ( X ) → π n ( f ) π n ( Y ) → π n − 1 ( F f ) . {\displaystyle \pi _{n}(X)\mathrel {\overset {\pi _{n}(f)}{\to }} \pi _{n}(Y)\to \pi _{n-1}(Ff).}

If the group on the right  π n − 1 ( F f ) {\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  x 0 ↪ X {\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

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This is instructive for a subset: an n-connected inclusion  A ↪ X {\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  A ↪ X {\displaystyle A\hookrightarrow X} to be 1-connected, it must be:

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

One-to-one on  π 0 ( A ) → π 0 ( X ) {\displaystyle \pi _{0}(A)\to \pi _{0}(X)} means that if there is a path connecting two points  a , b ∈ A {\displaystyle a,b\in A} by passing through X, there is a path in A connecting them, while onto  π 1 ( X ) {\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  π n − 1 ( A ) → π n − 1 ( X ) {\displaystyle \pi _{n-1}(A)\to \pi _{n-1}(X)} only implies that any elements of  π n − 1 ( A ) {\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  π n ( X ) {\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.

## Lower bounds

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

### Homology

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

The inequality may be strict: there are spaces in which  conn π ( X ) = 0 {\displaystyle {\text{conn}}_{\pi }(X)=0} but  conn H ( X ) = ∞ {\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

### Join

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

The identity is simpler with the eta notation:  η π ( K ∗ L ) ≥ η π ( K ) + η π ( L ) . {\displaystyle \eta _{\pi }(K*L)\geq \eta _{\pi }(K)+\eta _{\pi }(L).} As an example, let  K = L = S 0 = {\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  K ∗ L {\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  S 2 {\displaystyle S^{2}} , and its eta is 3. In general, the join of n copies of  S 0 {\displaystyle S^{0}} is homeomorphic to  S n − 1 {\displaystyle S^{n-1}} and its eta is n.

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

### Nerve

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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  J ⊂ I {\displaystyle 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.[7]: Thm.6

## Homotopy principle

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

## See also

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

## References

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1. ^ a b c d e f Matoušek, Jiří (2007). Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry (2nd ed.). Berlin-Heidelberg: Springer-Verlag. ISBN 978-3-540-00362-5. "Written in cooperation with Anders Björner and Günter M. Ziegler" , Section 4.3
2. ^ Aharoni, Ron; Berger, Eli (2006). "The intersection of a matroid and a simplicial complex". Transactions of the American Mathematical Society. 358 (11): 4895–4917\. doi:10.1090/S0002-9947-06-03833-5. ISSN 0002-9947.
3. ^ "n-connected space in nLab". ncatlab.org. Retrieved 2017-09-18.
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". Journal of Combinatorial Theory. Series A. 102 (1): 88–93\. doi:10.1016/S0097-3165(03)00015-3. ISSN 0097-3165.