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,
In general, for every integer d, (and ) [1]: 79, Thm.4.3.2 The proof requires two directions:
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: where 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 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:
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
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 is n-connected if and only if:
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:
If the group on the right vanishes, then the map on the left is a surjection.
Low-dimensional examples:
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 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 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 to be 1-connected, it must be:
One-to-one on means that if there is a path connecting two points by passing through X, there is a path in A connecting them, while onto 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 only implies that any elements of 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 ) 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.
Hurewicz theorem relates the homotopical connectivity to the homological connectivity, denoted by . 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, . Let ; so for all , and . Hurewicz theorem [5]: 366, Thm.4.32 says that, in this case, for all , and is isomorphic to , so too. Therefore:If X is not simply-connected (), thenstill holds. When this is trivial. When (so X is path-connected but not simply-connected), one should prove that .[ clarification needed]
The inequality may be strict: there are spaces in which but . [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 . Then: [1]: 81, Prop.4.4.3
The identity is simpler with the eta notation: As an example, let a set of two disconnected points. There is a 1-dimensional hole between the points, so the eta is 1. The join 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 , and its eta is 3. In general, the join of n copies of is homeomorphic to and its eta is n.
The general proof is based on a similar formula for the homological connectivity.
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 , the intersection 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 into a more general topological space, such as the space of all continuous maps between two associated spaces 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.
Written in cooperation with Anders Björner and Günter M. Ziegler, Section 4.3
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,
In general, for every integer d, (and ) [1]: 79, Thm.4.3.2 The proof requires two directions:
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: where 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 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:
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
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 is n-connected if and only if:
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:
If the group on the right vanishes, then the map on the left is a surjection.
Low-dimensional examples:
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 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 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 to be 1-connected, it must be:
One-to-one on means that if there is a path connecting two points by passing through X, there is a path in A connecting them, while onto 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 only implies that any elements of 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 ) 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.
Hurewicz theorem relates the homotopical connectivity to the homological connectivity, denoted by . 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, . Let ; so for all , and . Hurewicz theorem [5]: 366, Thm.4.32 says that, in this case, for all , and is isomorphic to , so too. Therefore:If X is not simply-connected (), thenstill holds. When this is trivial. When (so X is path-connected but not simply-connected), one should prove that .[ clarification needed]
The inequality may be strict: there are spaces in which but . [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 . Then: [1]: 81, Prop.4.4.3
The identity is simpler with the eta notation: As an example, let a set of two disconnected points. There is a 1-dimensional hole between the points, so the eta is 1. The join 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 , and its eta is 3. In general, the join of n copies of is homeomorphic to and its eta is n.
The general proof is based on a similar formula for the homological connectivity.
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 , the intersection 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 into a more general topological space, such as the space of all continuous maps between two associated spaces 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.
Written in cooperation with Anders Björner and Günter M. Ziegler, Section 4.3