Contents

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 $f \colon X \to Y$ is $n$-connected (or an $n$-equivalence) if for all $i \le n$ and all commutative squares

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

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

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

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

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 $B \sqcup_A^h C \to B' \sqcup_{A'}^h C'$ is $n$-connected.

This is (tom Dieck, Theorem 6.7.9).

Proposition

Let

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

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 $Y \times_X^h Z \to Y' \times_{X'}^h Z'$ is $n$-connected.

References

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

Revised on September 7, 2012 02:34:43 by Toby Bartels (98.23.143.147)