Homotopical connectivity

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.{{Cite Matousek 2007}}, Section 4.3{{Rp|page=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 $f\_d:\; S^d\; \backslash to\; X$.
• A d-dimensional ball in X is a continuous function $g\_d:\; B^d\; \backslash to\; X$.
• 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.{{Rp|page=78|location=Sec.4.3}}
• The homotopical connectivity of X, denoted $\backslash text\{conn\}\_\{\backslash pi\}(X)$, 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 $\backslash eta\_\{\backslash pi\}(X)$, and it differs from the previous parameter by 2, that is, $\backslash eta\_\{\backslash pi\}(X)\; :=\; \backslash text\{conn\}\_\{\backslash pi\}(X)\; +\; 2$.{{Cite journal |last1=Aharoni |first1=Ron |last2=Berger |first2=Eli |date=2006 |title=The intersection of a matroid and a simplicial complex |url=https://www.ams.org/tran/2006-358-11/S0002-9947-06-03833-5/ |journal=Transactions of the American Mathematical Society |language=en |volume=358 |issue=11 |pages=4895–4917 |doi=10.1090/S0002-9947-06-03833-5 |issn=0002-9947|doi-access=free }}

File:Hole_with_1-dimensional_boundary.png

• 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, $\backslash mathbb\{R\}^2\backslash setminus\; \backslash \{(0,0)\backslash \}$. 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 $\backslash text\{conn\}\_\{\backslash pi\}(X)\; =\; 0$. The lowest dimension of a hole is 2, so $\backslash eta\_\{\backslash pi\}(X)\; =\; 2$.File:Hole_with_a_2-dimensional_boundary.png
• 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 $\backslash text\{conn\}\_\{\backslash pi\}(X)\; =\; 1$. The smallest dimension of a hole is 3, so $\backslash eta\_\{\backslash pi\}(X)\; =\; 3$.

File:Hole_with_a_0-dimensional_boundary.png

• 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 $\backslash text\{conn\}\_\{\backslash pi\}(X)\; =\; -1$. The lowest dimension of a hole is 1, so $\backslash eta\_\{\backslash pi\}(X)\; =\; 1$.
• 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, $\backslash text\{conn\}\_\{\backslash pi\}(X)\; =\; -2$ and $\backslash eta\_\{\backslash pi\}(X)\; =\; 0$, which is its smallest possible value.
• A ball has no holes of any dimension. Therefore, its connectivity is infinite: $\backslash eta\_\{\backslash pi\}(X)\; =\; \backslash text\{conn\}\_\{\backslash pi\}(X)\; =\; \backslash infty$.

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

• Proving that , that is, $S^d$ cannot be continuously shrunk to a single point. This can be proved using the Borsuk–Ulam theorem.
• Proving that $\backslash text\{conn\}\_\{\backslash pi\}(S^d)\; \backslash geq\; d-1$, that is, that is, every continuous map $S^k\; \backslash to\; S^d$ for 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: $$\backslash pi\_d(X)\; \backslash cong\; 0,\; \backslash quad\; -1\; \backslash leq\; d\; \backslash leq\; n,$$ where $\backslash pi\_i(X)$ denotes the i-th homotopy group and 0 denotes the trivial group.{{Cite web |title=n-connected space in nLab |url=https://ncatlab.org/nlab/show/n-connected+space |access-date=2017-09-18 |website=ncatlab.org}} 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 $\backslash pi\_d(X)\; \backslash not\; \backslash cong\; 0$.The homotopical connectivity of X is the largest integer n for which X is n-connected.{{cite arXiv |last1=Frick |first1=Florian |last2=Soberón |first2=Pablo |date=2020-05-11 |title=The topological Tverberg problem beyond prime powers |class=math.CO |eprint=2005.05251 }}

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:

:$\backslash pi\_0(X,\; *)\; :=\; \backslash left[\backslash left(S^0,\; *\backslash right),\; \backslash left(X,\; *\backslash right)\backslash right].$

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

:$\backslash pi\_i(X)\; \backslash simeq\; 0,\; \backslash quad\; 0\; \backslash leq\; i\; \backslash leq\; n.$

• 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 $f\backslash colon\; X\; \backslash to\; Y$ is n-connected if and only if:

• $\backslash pi\_i(f)\backslash colon\; \backslash pi\_i(X)\; \backslash mathrel\{\backslash overset\{\backslash sim\}\{\backslash to\}\}\; \backslash pi\_i(Y)$ is an isomorphism for , and
• $\backslash pi\_n(f)\backslash colon\; \backslash pi\_n(X)\; \backslash twoheadrightarrow\; \backslash pi\_n(Y)$ 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

:$\backslash pi\_n(X)\; \backslash mathrel\{\backslash overset\{\backslash pi\_n(f)\}\{\backslash to\}\}\; \backslash pi\_n(Y)\; \backslash to\; \backslash pi\_\{n-1\}(Ff).$

If the group on the right $\backslash pi\_\{n-1\}(Ff)$ 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 $x\_0\; \backslash hookrightarrow\; X$ 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 $A\; \backslash hookrightarrow\; X$ 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 $A\; \backslash hookrightarrow\; X$ to be 1-connected, it must be:

• onto $\backslash pi\_0(X),$
• one-to-one on $\backslash pi\_0(A)\; \backslash to\; \backslash pi\_0(X),$ and
• onto $\backslash pi\_1(X).$

One-to-one on $\backslash pi\_0(A)\; \backslash to\; \backslash pi\_0(X)$ means that if there is a path connecting two points $a,\; b\; \backslash in\; A$ by passing through X, there is a path in A connecting them, while onto $\backslash pi\_1(X)$ 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 $\backslash pi\_\{n-1\}(A)\; \backslash to\; \backslash pi\_\{n-1\}(X)$ only implies that any elements of $\backslash pi\_\{n-1\}(A)$ 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 $\backslash pi\_n(X)$) 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 $\backslash text\{conn\}\_\{\backslash pi\}(X)$ to the homological connectivity, denoted by $\backslash text\{conn\}\_H(X)$. 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, $\backslash text\{conn\}\_\{\backslash pi\}(X)\backslash geq\; 1$. Let $n\; :=\; \backslash text\{conn\}\_\{\backslash pi\}(X)\; +\; 1\backslash geq\; 2$; so $\backslash pi\_i(X)=\; 0$ for all X is not simply-connected ($\backslash text\{conn\}\_\{\backslash pi\}(X)\backslash leq\; 0$), then$$\backslash text\{conn\}\_H(X)\backslash geq\; \backslash text\{conn\}\_\{\backslash pi\}(X)$$still holds. When $\backslash text\{conn\}\_\{\backslash pi\}(X)\backslash leq-1$ this is trivial. When $\backslash text\{conn\}\_\{\backslash pi\}(X)=0$ (so X is path-connected but not simply-connected), one should prove that $\backslash tilde\{H\_0\}(X)=\; 0$.{{Clarify|date=May 2022}}

The inequality may be strict: there are spaces in which $\backslash text\{conn\}\_\{\backslash pi\}(X)=0$ but $\backslash text\{conn\}\_H(X)=\backslash infty$.See example 2.38 in Hatcher's book. See also [https://math.stackexchange.com/a/4441587/29780 this answer].

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.{{Rp|page=80|location=Prop.4.4.2}}

Let K and L be non-empty cell complexes. Their join is commonly denoted by $K\; *\; L$. Then:''{{Rp|page=81|location=Prop.4.4.3}}

$$\backslash text\{conn\}\_\{\backslash pi\}(K*L)\; \backslash geq\; \backslash text\{conn\}\_\{\backslash pi\}(K)+\backslash text\{conn\}\_\{\backslash pi\}(L)+2.$$

The identity is simpler with the eta notation:

$$\backslash eta\_\{\backslash pi\}(K*L)\; \backslash geq\; \backslash eta\_\{\backslash pi\}(K)+\backslash eta\_\{\backslash pi\}(L).$$

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.

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 $J\backslash subset\; I$, the intersection $\backslash bigcap\_\{i\backslash in\; J\}\; U\_i$ 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.{{Cite journal |last=Björner |first=Anders |authorlink1=Anders Björner |date=2003-04-01 |title=Nerves, fibers and homotopy groups |journal=Journal of Combinatorial Theory | series=Series A |language=en |volume=102 |issue=1 |pages=88–93 |doi=10.1016/S0097-3165(03)00015-3 |doi-access=free |issn=0097-3165}}{{Rp|location=Thm.6}}

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

{{reflist}}

Category:General topology

Category:Properties of topological spaces

Category:Homotopy theory