6
$\begingroup$

I want to show the following:

$X$ $n$-connected $\iff $ any continuous map $f:K \rightarrow X$ where $K$ is a cell complex of dimension $\leq n$ is homotopic to a constant map

For this I think I can use the following: $X$ $n$-connected $\iff $ every continuous map $f: S^n \rightarrow X$ is homotopic to a constant map.

Proof:

"$\Leftarrow$"

If any continuous map $f:K \rightarrow X$ where $K$ is a cell complex of dimension $\leq n$ is homotopic to a constant map then any $f: S^n \rightarrow X$ is homotopic to a constant map. So $X$ is $n$-connected.

"$\Rightarrow$"

I'm not sure how to proceed in this direction. I know $X$ is $n$-connected and so $\pi_i (X) = 0$ for all $i \leq n$. I also know any $f: S^i \rightarrow X$ is null-homotopic.

How to proceed from here? Many thanks for your help!

  • 0
    @Shaun Ault: But $X$ is not necessarily a CW complex. It's just a topological space.2011-09-08

1 Answers 1

2

This is an application of the second theorem of chapter 10.3 in May: A Concise Course in Algebraic Topology, i.e. there you can find your proof.