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.

Not to be confused with Homotopic connectivity.

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".

A d-dimensional sphere in X is a $f_{d}:S^{d}\to X$ .

continuous function

A d-dimensional ball in X is a continuous function $g_{d}:B^{d}\to X$ .

A d-dimensional-boundary hole in X is a d-dimensional sphere that is not (- 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").

nullhomotopic

X is called n-connected if it contains no holes of boundary-dimension d ≤ n.^{: 78, Sec.4.3}

[1]

The homotopical connectivity of X, denoted ${\text{conn}}_{\pi }(X)$ , 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 $\eta _{\pi }(X)$ , and it differs from the previous parameter by 2, that is, $\eta _{\pi }(X):={\text{conn}}_{\pi }(X)+2$ .

[2]

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 $\pi _{d}(X)\not \cong 0$ .The homotopical connectivity of X is the largest integer n for which X is n-connected.

[4]

$\pi _{i}(f)\colon \pi _{i}(X)\mathrel {\overset {\sim }{\to }} \pi _{i}(Y)$ is an isomorphism for $i<n$ , and

$\pi _{n}(f)\colon \pi _{n}(X)\twoheadrightarrow \pi _{n}(Y)$ is a surjection.

Homotopy principle[edit]

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

Homotopical connectivity

continuous function

nullhomotopic

[1]

[2]

[4]

Homotopy principle[edit]

Connected space

Connective spectrum

Path-connected

Simply connected