0
$\begingroup$

I've been given the question:

Let $ f : X \to Y $ be a continuous map, and suppose we are given (not necessarily equal) continuous maps $ g,h : Y \to X $ such that $ gf \simeq id_X $ and $ fh \simeq id_Y $. Show that $f$ is a homotopy equivalence.

What does it mean for a single function $ f: X \to Y $ to be a homotopy equivalence?

Thanks

$\endgroup$
  • 4
    $\begingroup$ It means there is a function $j:Y\rightarrow X$ so that $jf\simeq id_X$ and $fj\simeq id_Y$. $\endgroup$ – Jason DeVito Oct 13 '11 at 16:05
1
$\begingroup$

Is there a specific part of the definition you don't understand? From Wikipedia:

Given two spaces $X$ and $Y$, we say they are homotopy equivalent or of the same homotopy type if there exist continuous maps $f : X → Y$ and $g : Y → X$ such that $g ∘ f$ is homotopic to the identity map id $X$ and $f ∘ g$ is homotopic to id $Y$.

The article also gives a definition of homotopy between functions

Formally, a homotopy between two continuous functions $f$ and $g$ from a topological space $X$ to a topological space $Y$ is defined to be a continuous function $H : X × [0,1] → Y$ from the product of the space X with the unit interval $[0,1]$ to $Y$ such that, if $x ∈ X$ then $H(x,0) = f(x)$ and $H(x,1) = g(x)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.