nLab
n-connected continuous function

Contents

Idea

In algebraic topology, a continuous function between topological spaces is called nn-connected if it induces isomorphisms on all homotopy groups in degree <n\lt n and epimorphisms in degree nn. In older literature this is often called an nn-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 nn-connected function represents an n-connected morphism in the (∞,1)-topos ∞Gpd.

Definition

Definition

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

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

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

    2. an epimorphism in degree nn.

  2. for all ini \le n and all commutative squares

S i1 u X f D i v Y \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 iXw \colon D^i \to X such that w|S i1=uw | S^{i-1} = u and fwf w is homotopic to vv relative to S i1S^{i-1}.

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

Properties

Proposition

For a map f:XYf \colon X \to Y and an integer n1n \ge -1 the following conditions are equivalent.

  1. ff is nn-connected.

  2. All homotopy fibers of ff are (n1)(n-1)-connected.

Proposition

Let f:XYf \colon X \to Y and g:YZg \colon Y \to Z be maps of spaces.

  1. If ff and gg are nn-connected, then so is gfg f.

  2. If ff is (n1)(n-1)-connected and gfg f is nn-connected, then gg is nn-connected.

  3. If gg is (n+1)(n+1)-connected and gfg f is nn-connected, then ff is nn-connected.

Proposition

Let

B A C g f h B A C \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 ff is (n1)(n-1)-connected and gg and hh are nn-connected, then the induced map between homotopy pushouts B A hCB A hCB \sqcup_A^h C \to B' \sqcup_{A'}^h C' is nn-connected.

This is (tom Dieck, Theorem 6.7.9).

Proposition

Let

Y X Z g f h Y X Z \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 ff is (n+1)(n+1)-connected and gg and hh are nn-connected, then the induced map between homotopy pullbacks Y× X hZY× X hZY \times_X^h Z \to Y' \times_{X'}^h Z' is nn-connected.

References

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

Revised on May 4, 2016 13:35:56 by Urs Schreiber (131.220.184.222)