nLab
n-connected map

Definition

An $n$-connected map is an $n$-connected morphism in the (∞,1)-topos ∞Gpd, usually considered as presented by the model category Top of topological spaces.

Definition

A map of topological spaces $f:X\to Y$ is $n$-connected (or an $n$-equivalence) if for all $i\le n$ and all commutative squares

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

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

Properties

Proposition

For a map $f:X\to Y$ 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 $f:X\to Y$ and $g:Y\to Z$ be maps of spaces.

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

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

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

Proposition

Let

$$\begin{array}{ccccc}B& ⟵& A& ⟶& C\\ g\downarrow & & \downarrow f& & \downarrow h\\ B\prime & ⟵& A\prime & ⟶& C\prime \end{array}$$\begin{matrix} B & \longleftarrow & A & \longrightarrow & C \\ g \downarrow & & \downarrow f & & \, \downarrow 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 $B{\bigsqcup }_A^{h}C\to B\prime {\bigsqcup }_{A\prime }^{h}C\prime$ is $n$-connected.

This is (tom Dieck, Theorem 6.7.9).

Proposition

Let

$$\begin{array}{ccccc}Y& ⟶& X& ⟵& Z\\ g\downarrow & & \downarrow f& & \downarrow h\\ Y\prime & ⟶& X\prime & ⟵& Z\prime \end{array}$$\begin{matrix} Y & \longrightarrow & X & \longleftarrow & Z \\ g \downarrow & & \downarrow f & & \, \downarrow 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 $Y{\times }_X^{h}Z\to Y\prime {\times }_{X\prime }^{h}Z\prime$ is $n$-connected.

References

• Tammo tom Dieck, Algebraic topology. European Mathematical Society, Zürich, 2008.