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Let's define the following:

(Def) A continuous map $f_0 : A \to B$ where $A,B$ are topological spaces is called essential if every homotopic map $f_1$ is surjective, i.e. $f_1 (A) = B$.

Let $B=S^n$ be the $n$-sphere and let $f_0 : A \to B$ be non-essential. Claim: Then there exists a homotopic map $f_1$ such that $f_1 (A) = \ast$ is a one point space.

That's a footnote in the paper I'm reading. But I don't understand the claim. If $A$ is simply-connected then I think I "see" that $f_0(A) = f_1(A) = \ast$. But the claim in the paper does not put any restrictions on $A$ so in particular, it could be disconnected. I'm a bit confused. Would someone show me how the claim is true? Thanks lots.

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Rudy the Reindeer
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    Inflate the point you miss and contract the rest of the image to the antipode. – t.b. Sep 07 '12 at 13:12
  • @t.b. But what if it looks like a bunch of scattered points? How do I contract that? – Rudy the Reindeer Sep 07 '12 at 13:15
  • Wait. What? I want to continuously retract the image to a point, right? – Rudy the Reindeer Sep 07 '12 at 13:16
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    I think you are confusing the idea of homotopic maps with homotopic spaces. Consider the map $f: \lbrace 0,1 \rbrace \to \mathbb{R}$ such that $f(0)=0$ and $f(1)=1$. Then we can make a homotopy $H:\lbrace 0,1\rbrace \times [0,1] \to \mathbb{R}$ defined by $H(x,t)=xt$. Then $H(x,1)=f$ and $H(x,0)=0$. Thus while the image of $f$ is disconnected. The image of $H(x,0)$ is a one point space. – Kris Williams Sep 07 '12 at 13:53
  • @DrKW Thank you very much for this comment. I did indeed confuse the images of homotopic maps with homotopic spaces. – Rudy the Reindeer Sep 07 '12 at 14:21

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You confused homotopic maps with homotopic spaces, as pointed out by DrKW in the comments. Here is what you can do to the image if $f$ is not surjective:

enter image description here

This does not work if $f$ is surjective since then you'd have to "tear" the image. But of course something that "tears" your space cannot be continuous.

Rudy the Reindeer
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