nLab n-connected continuous function

Idea

In algebraic topology, a continuous function between topological spaces is called $n$-connected if it induces isomorphisms on all homotopy groups in degree $\lt n$ and epimorphisms in degree $n$.

In older literature this is often called an $n$-equivalence, since an $\infty$-equivalence in this sense is a weak homotopy equivalence.

In terms of the homotopy theory presented by the classical model structure on topological spaces, an $n$-connected function represents an n-connected morphism in the (∞,1)-topos ∞Gpd.

Definition

Definition

A map of topological spaces $f \colon X \to Y$ is called $n$-connected (e.g. tomDieck 08, p. 144) or an $n$-equivalence (older literature) if the following equivalent definitions hold:

1. The induced morphism on homotopy groups $\pi_\bullet(X,x)\to \pi_\bullet(Y,f(x))$ is, for all $x\in X$

   1. an isomorphism in degree $\lt n$;

   2. an epimorphism in degree $n$.

2. for all $i \le n$ and all commutative squares

$$\begin{matrix} S^{i-1} & \overset{u}{\longrightarrow} & X \\ \big\downarrow & & \, \big\downarrow^\mathrlap{f} \\ D^i & \underset{v}{\longrightarrow} & Y \end{matrix}$$

there exists a map $w \colon D^i \to X$ such that $w | S^{i-1} = u$ and $f w$ is homotopic to $v$ relative to $S^{i-1}$.

Hence an $\infty$-connected map is a weak homotopy equivalence.

Properties

Proposition

For a map $f \colon X \to Y$ and an integer $n \ge -1$ the following conditions are equivalent.

1. $f$ is $n$-connected.

2. All homotopy fibers of $f$ are $(n-1)$-connected.

Proposition

Let $f \colon X \to Y$ and $g \colon Y \to Z$ be maps of spaces.

1. If $f$ and $g$ are $n$-connected, then so is $g f$.

2. If $f$ is $(n-1)$-connected and $g f$ is $n$-connected, then $g$ is $n$-connected.

3. If $g$ is $(n+1)$-connected and $g f$ is $n$-connected, then $f$ is $n$-connected.

Proposition

Let

$$\begin{matrix} B & \longleftarrow & A & \longrightarrow & C \\ \mathllap{{}^g}\big\downarrow & & \big\downarrow^\mathrlap{f} & & \, \big\downarrow^\mathrlap{h} \\ B' & \longleftarrow & A' & \longrightarrow & C' \end{matrix}$$

be a commutative diagram of maps of spaces. If $f$ is $(n-1)$-connected and $g$ and $h$ are $n$-connected, then the induced map between homotopy pushouts $B \sqcup_A^h C \to B' \sqcup_{A'}^h C'$ is $n$-connected.

This is (tom Dieck, Theorem 6.7.9).

Proposition

Let

$$\begin{matrix} Y & \longrightarrow & X & \longleftarrow & Z \\ \mathllap{{}^g} \big\downarrow & & \big\downarrow^\mathrlap{f} & & \, \big\downarrow^\mathrlap{h} \\ Y' & \longrightarrow & X' & \longleftarrow & Z' \end{matrix}$$

be a commutative diagram of maps of spaces. If $f$ is $(n+1)$-connected and $g$ and $h$ are $n$-connected, then the induced map between homotopy pullbacks $Y \times_X^h Z \to Y' \times_{X'}^h Z'$ is $n$-connected.

Related concepts

• Postnikov tower

• n-image

References

Textbook accounts:

• Tammo tom Dieck, p. 144 and Thm 6.7.9 in: Algebraic topology, European Mathematical Society, Zürich (2008) (doi:10.4171/048, pdf)